Ring (mathematics)
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In , rings are s that generalize : multiplication need not be and s need not exist. In other words, a ''ring'' is a equipped with two s satisfying properties analogous to those of and of s. Ring elements may be numbers such as s or s, but they may also be non-numerical objects such as s, , , and . Formally, a ''ring'' is an whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is , is over the addition operation, and has a multiplicative . (Some authors use the term "ring" to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. , the theory of s, is a major branch of . Its development has been greatly influenced by problems and ideas of and . The simplest commutative rings are those that admit division by non-zero elements; such rings are called . Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the of an , and the of a number field. Examples of noncommutative rings include the ring of real with , s in , s in , , and s in . The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by , , , and . Rings were first formalized as a generalization of s that occur in , and of s and rings of invariants that occur in and . They later proved useful in other branches of mathematics such as and .


Definition

A ring is a ''R'' equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms # ''R'' is an under addition, meaning that: #* (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') for all ''a'', ''b'', ''c'' in ''R''   (that is, + is ). #* ''a'' + ''b'' = ''b'' + ''a'' for all ''a'', ''b'' in ''R''   (that is, + is ). #* There is an element 0 in ''R'' such that ''a'' + 0 = ''a'' for all ''a'' in ''R''   (that is, 0 is the ). #* For each ''a'' in ''R'' there exists −''a'' in ''R'' such that ''a'' + (−''a'') = 0   (that is, −''a'' is the of ''a''). # ''R'' is a under multiplication, meaning that: #* (''a'' ⋅ ''b'') ⋅ ''c'' = ''a'' ⋅ (''b'' ⋅ ''c'') for all ''a'', ''b'', ''c'' in ''R''   (that is, ⋅ is associative). #* There is an element 1 in ''R'' such that and for all ''a'' in ''R''   (that is, 1 is the ). # Multiplication is with respect to addition, meaning that: #* ''a'' ⋅ (''b'' + ''c'') = (''a'' ⋅ ''b'') + (''a'' ⋅ ''c'') for all ''a'', ''b'', ''c'' in ''R''   (left distributivity). #* (''b'' + ''c'') ⋅ ''a'' = (''b'' ⋅ ''a'') + (''c'' ⋅ ''a'') for all ''a'', ''b'', ''c'' in ''R''   (right distributivity).


Notes on the definition

In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a (IPA: ). For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in ' below, many authors apply the term "ring" without requiring a multiplicative identity. The multiplication symbol ⋅ is usually omitted; for example, ''xy'' means . Although ring addition is , ring multiplication is not required to be commutative: ''ab'' need not necessarily equal ''ba''. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called ''s''. Books on commutative algebra or algebraic geometry often adopt the convention that ''ring'' means ''commutative ring'', to simplify terminology. In a ring, multiplicative inverses are not required to exist. A non commutative ring in which every nonzero element has a is called a . The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferrable from the remaining rng assumptions only for elements that are products: .) Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every is a "ring".


Illustration

The most familiar example of a ring is the set of all integers \mathbf, consisting of the s : ... , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.


Some properties

Some basic properties of a ring follow immediately from the axioms: * The additive identity is unique. * The additive inverse of each element is unique. * The multiplicative identity is unique. * For any element ''x'' in a ring ''R'', one has (zero is an with respect to multiplication) and . * If in a ring ''R'' (or more generally, 0 is a unit element), then ''R'' has only one element, and is called the . * If a ring ''R'' contains the zero ring as a subring, then ''R'' itself is the zero ring. * The holds for any ''x'' and ''y'' satisfying .


Example: Integers modulo 4

Equip the set \mathbf/4\mathbf = \left\ with the following operations: * The sum \overline + \overline in Z/4Z is the remainder when the integer is divided by 4 (as is always smaller than 8, this remainder is either or ). For example, \overline + \overline = \overline and \overline + \overline = \overline. * The product \overline \cdot \overline in Z/4Z is the remainder when the integer ''xy'' is divided by 4. For example, \overline \cdot \overline = \overline and \overline \cdot \overline = \overline. Then Z/4Z is a ring: each axiom follows from the corresponding axiom for Z. If ''x'' is an integer, the remainder of ''x'' when divided by 4 may be considered as an element of Z/4Z, and this element is often denoted by or \overline, which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any \overline in Z/4Z is \overline. For example, -\overline = \overline = \overline.


Example: 2-by-2 matrices

The set of 2-by-2 with entries in a is :\operatorname_2(F) = \left\. With the operations of matrix addition and , \operatorname_2(F) satisfies the above ring axioms. The element \left( \begin 1 & 0 \\ 0 & 1 \end\right) is the multiplicative identity of the ring. If A = \left( \begin 0 & 1 \\ 1 & 0 \end \right) and B = \left( \begin 0 & 1 \\ 0 & 0 \end \right), then AB = \left( \begin 0 & 0 \\ 0 & 1 \end \right) while BA = \left( \begin 1 & 0 \\ 0 & 0 \end \right); this example shows that the ring is noncommutative. More generally, for any ring , commutative or not, and any nonnegative integer , the square matrices of dimension with entries in form a ring: see .


History


Dedekind

The study of rings originated from the theory of s and the theory of s.
/ref> In 1871, defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by 's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.


Hilbert

The term "Zahlring" (number ring) was coined by in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an ). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if then , , , , , and so on; in general, ''a''''n'' is going to be an integral linear combination of 1, ''a'', and ''a''2.


Fraenkel and Noether

The first axiomatic definition of a ring was given by in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every to have a . In 1921, gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper ''Idealtheorie in Ringbereichen''.


Multiplicative identity and the term "ring"

Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas Noether's did not. Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2006 that use the term without the requirement for a 1. Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." makes the counterargument that rings without a multiplicative identity are not totally associative (the product of any finite sequence of ring elements, including the empty sequence, is well-defined, independent of the order of operations) and writes "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1". Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention: :* to include a requirement a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit", or "ring with 1". :* to omit a requirement for a multiplicative identity: "rng" or "pseudo-ring", although the latter may be confusing because it also has other meanings.


Basic examples


Commutative rings

* The prototypical example is the ring of integers with the two operations of addition and multiplication. * The rational, real and complex numbers are commutative rings of a type called . * A unital associative is itself a ring as well as an . Some examples: ** The algebra of with coefficients in . As an -module, is of infinite rank. ** The algebra of with coefficients in . ** The set of all real-valued defined on the real line forms a commutative -algebra. The operations are addition and multiplication of functions. ** Let be a set, and let be a ring. Then the set of all functions from to forms a ring, which is commutative if is commutative. The ring of continuous functions in the previous example is a subring of this ring if is the real line and . * \mathbf /math>, the integers with a real or complex number adjoined. As a -module, it is free of infinite rank if is , free of finite rank if is an algebraic integer, and not free otherwise. * \mathbf /10/math>, the set of s. Not free as a -module. * \mathbf\left left(1 + \sqrt\right)/2\right/math>, where is a integer of the form , with . A free -module of rank 2. See '. * \mathbf /math>, the s. * \mathbf left(1 + \sqrt\right)/2/math>, the s. * The previous two examples are the cases and of the . * The previous four examples are cases of the of a , defined as the set of s in . * The set of all algebraic integers in forms a ring called the of in . * If is a set, then the of becomes a ring if we define addition to be the of sets and multiplication to be . This is an example of a .


Noncommutative rings

* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n'' with entries from ''R'', forms a ring with matrix addition and matrix multiplication as operations. For , this matrix ring is isomorphic to ''R'' itself. For (and ''R'' not the zero ring), this matrix ring is noncommutative. * If ''G'' is an , then the of ''G'' form a ring, the End(''G'') of ''G''. The operations in this ring are addition and composition of endomorphisms. More generally, if ''V'' is a over a ring ''R'', then the set of all ''R''-linear maps forms a ring, also called the endomorphism ring and denoted by End''R''(''V''). * If ''G'' is a and ''R'' is a ring, the of ''G'' over ''R'' is a over ''R'' having ''G'' as basis. Multiplication is defined by the rules that the elements of ''G'' commute with the elements of ''R'' and multiply together as they do in the group ''G''. * Many rings that appear in analysis are noncommutative. For example, most s are noncommutative.


Non-rings


Basic concepts


Products and powers

For each nonnegative integer , given a sequence (a_1,\ldots,a_n) of elements of , one can define the product \textstyle P_n = \prod_^n a_i recursively: let and let for . As a special case, one can define nonnegative integer powers of an element of a ring: and for . Then for all .


Elements in a ring

A left of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0. A right zero divisor is defined similarly. A is an element a such that a^n = 0 for some n > 0. One example of a nilpotent element is a . A nilpotent element in a is necessarily a zero divisor. An e is an element such that e^2 = e. One example of an idempotent element is a in linear algebra. A is an element a having a ; in this case the inverse is unique, and is denoted by a^. The set of units of a ring is a under ring multiplication; this group is denoted by R^\times or R^* or U(R). For example, if ''R'' is the ring of all square matrices of size ''n'' over a field, then R^\times consists of the set of all invertible matrices of size ''n'', and is called the .


Subring

A subset ''S'' of ''R'' is called a if any one of the following equivalent conditions holds: * the addition and multiplication of ''R'' to give operations ''S'' × ''S'' → ''S'' making ''S'' a ring with the same multiplicative identity as ''R''. * 1 ∈ ''S''; and for all ''x'', ''y'' in ''S'', the elements ''xy'', ''x'' + ''y'', and −''x'' are in ''S''. * ''S'' can be equipped with operations making it a ring such that the inclusion map ''S'' → ''R'' is a ring homomorphism. For example, the ring Z of integers is a subring of the of real numbers and also a subring of the ring of s Z 'X''(in both cases, Z contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2Z does not contain the identity element 1 and thus does not qualify as a subring of Z; one could call 2Z a , however. An intersection of subrings is a subring. Given a subset ''E'' of ''R'', the smallest subring of ''R'' containing ''E'' is the intersection of all subrings of ''R'' containing ''E'', and it is called ''the subring generated by E''. For a ring ''R'', the smallest subring of ''R'' is called the ''characteristic subring'' of ''R''. It can be generated through addition of copies of 1 and −1. It is possible that n\cdot 1=1+1+\ldots+1 (''n'' times) can be zero. If ''n'' is the smallest positive integer such that this occurs, then ''n'' is called the ' of ''R''. In some rings, n\cdot 1 is never zero for any positive integer ''n'', and those rings are said to have ''characteristic zero''. Given a ring ''R'', let \operatorname(R) denote the set of all elements ''x'' in ''R'' such that ''x'' commutes with every element in ''R'': xy = yx for any ''y'' in ''R''. Then \operatorname(R) is a subring of ''R'', called the of ''R''. More generally, given a subset ''X'' of ''R'', let ''S'' be the set of all elements in ''R'' that commute with every element in ''X''. Then ''S'' is a subring of ''R'', called the (or commutant) of ''X''. The center is the centralizer of the entire ring ''R''. Elements or subsets of the center are said to be ''central'' in ''R''; they (each individually) generate a subring of the center.


Ideal

Let ''R'' be a ring. A left ideal of ''R'' is a nonempty subset ''I'' of ''R'' such that for any ''x'', ''y'' in ''I'' and ''r'' in ''R'', the elements x+y and rx are in ''I''. If R I denotes the ''R''-span of ''I'', that is, the set of finite sums :r_1 x_1 + \cdots + r_n x_n \quad \textrm\;\textrm\; r_i \in R \; \textrm \; x_i \in I, then ''I'' is a left ideal if R I \subseteq I. Similarly, a right ideal is a subset ''I'' such that I R \subseteq I. A subset ''I'' is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of ''R''. If ''E'' is a subset of ''R'', then R E is a left ideal, called the left ideal generated by ''E''; it is the smallest left ideal containing ''E''. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of ''R''. If ''x'' is in ''R'', then Rx and xR are left ideals and right ideals, respectively; they are called the left ideals and right ideals generated by ''x''. The principal ideal RxR is written as (x). For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field. Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite of left ideals is called a left . A ring in which there is no strictly decreasing infinite chain of left ideals is called a left . It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the ). The integers, however, form a Noetherian ring which is not Artinian. For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal ''P'' of ''R'' is called a if for any elements x, y\in R we have that xy \in P implies either x \in P or y\in P. Equivalently, ''P'' is prime if for any ideals I, J we have that IJ \subseteq P implies either I \subseteq P or J \subseteq P. This latter formulation illustrates the idea of ideals as generalizations of elements.


Homomorphism

A from a ring to a ring is a function ''f'' from ''R'' to ''S'' that preserves the ring operations; namely, such that, for all ''a'', ''b'' in ''R'' the following identities hold: * ''f''(''a'' + ''b'') = ''f''(''a'') ‡ ''f''(''b'') * ''f''(''a'' ⋅ ''b'') = ''f''(''a'') ∗ ''f''(''b'') * ''f''(1''R'') = 1''S'' If one is working with rngs, then the third condition is dropped. A ring homomorphism ''f'' is said to be an if there exists an inverse homomorphism to ''f'' (that is, a ring homomorphism that is an ). Any ring homomorphism is a ring isomorphism. Two rings R, S are said to be isomorphic if there is an isomorphism between them and in that case one writes R \simeq S. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism. Examples: * The function that maps each integer ''x'' to its remainder modulo 4 (a number in ) is a homomorphism from the ring Z to the quotient ring Z/4Z ("quotient ring" is defined below). * If u is a unit element in a ring ''R'', then R \to R, x \mapsto uxu^ is a ring homomorphism, called an of ''R''. * Let ''R'' be a commutative ring of prime characteristic ''p''. Then x \mapsto x^p is a ring endomorphism of ''R'' called the . * The of a field extension L/K is the set of all automorphisms of ''L'' whose restrictions to ''K'' are the identity. * For any ring ''R'', there are a unique ring homomorphism and a unique ring homomorphism . * An (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map is an epimorphism. * An algebra homomorphism from a ''k''-algebra to the of a vector space over ''k'' is called a . Given a ring homomorphism f:R \to S, the set of all elements mapped to 0 by ''f'' is called the of ''f''. The kernel is a two-sided ideal of ''R''. The image of ''f'', on the other hand, is not always an ideal, but it is always a subring of ''S''. To give a ring homomorphism from a commutative ring ''R'' to a ring ''A'' with image contained in the center of ''A'' is the same as to give a structure of an over ''R'' to ''A'' (which in particular gives a structure of an ''A''-module).


Quotient ring

The notion of is analogous to the notion of a . Given a ring and a two-sided ''I'' of , view ''I'' as subgroup of ; then the quotient ring ''R''/''I'' is the set of s of ''I'' together with the operations :(''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + ''I'' and :(''a'' + ''I'')(''b'' + ''I'') = (''ab'') + ''I''. for all ''a'', ''b'' in ''R''. The ring ''R''/''I'' is also called a factor ring. As with a quotient group, there is a canonical homomorphism p \colon R \to R/I, given by x \mapsto x + I. It is surjective and satisfies the following universal property: *If f \colon R \to S is a ring homomorphism such that f(I) = 0, then there is a unique homomorphism \overline \colon R/I \to S such that f = \overline \circ p. For any ring homomorphism f \colon R \to S, invoking the universal property with I = \ker f produces a homomorphism \overline \colon R/\ker f \to S that gives an isomorphism from R/\ker f to the image of .


Module

The concept of a ''module over a ring'' generalizes the concept of a (over a ) by generalizing from multiplication of vectors with elements of a field () to multiplication with elements of a ring. More precisely, given a ring with 1, an -module is an equipped with an (associating an element of to every pair of an element of and an element of ) that satisfies certain . This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all in and all in , we have: * is an abelian group under addition. * a(x+y)=ax+ay * (a+b)x=ax+bx * 1x=x * (ab)x=a(bx) When the ring is these axioms define ''left modules''; ''right modules'' are defined similarly by writing instead of . This is not only a change of notation, as the last axiom of right modules (that is ) becomes , if left multiplication (by ring elements) is used for a right module. Basic examples of modules are ideals, including the ring itself. Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the ). In particular, not all modules have a . The axioms of modules imply that , where the first minus denotes the in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers. Any ring homomorphism induces a structure of a module: if is a ring homomorphism, then is a left module over by the multiplication: . If is commutative or if is contained in the of , the ring is called a -. In particular, every ring is an algebra over the integers.


Constructions


Direct product

Let ''R'' and ''S'' be rings. Then the can be equipped with the following natural ring structure: * (''r''1, ''s''1) + (''r''2, ''s''2) = (''r''1 + ''r''2, ''s''1 + ''s''2) * (''r''1, ''s''1) ⋅ (''r''2, ''s''2) = (''r''1 ⋅ ''r''2, ''s''1 ⋅ ''s''2) for all ''r''1, ''r''2 in ''R'' and ''s''1, ''s''2 in ''S''. The ring with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the of ''R'' with ''S''. The same construction also works for an arbitrary family of rings: if R_i are rings indexed by a set ''I'', then \textstyle \prod_ R_i is a ring with componentwise addition and multiplication. Let ''R'' be a commutative ring and \mathfrak_1, \cdots, \mathfrak_n be ideals such that \mathfrak_i + \mathfrak_j = (1) whenever i \ne j. Then the says there is a canonical ring isomorphism: :R / \simeq \prod_^, \qquad x \;\operatorname\; \mapsto (x \;\operatorname\; \mathfrak_1, \ldots , x \;\operatorname\; \mathfrak_n). A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let R_i, 1 \le i \le n be rings, \textstyle R_i \to R = \prod R_i the inclusions with the images \mathfrak_i (in particular \mathfrak_i are rings though not subrings). Then \mathfrak_i are ideals of ''R'' and :R = \mathfrak_1 \oplus \cdots \oplus \mathfrak_n, \quad \mathfrak_i \mathfrak_j = 0, i \ne j, \quad \mathfrak_i^2 \subseteq \mathfrak_i as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to ''R''. Equivalently, the above can be done through s. Assume that ''R'' has the above decomposition. Then we can write :1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak_i. By the conditions on \mathfrak_i, one has that e_i are central idempotents and e_i e_j = 0, i \ne j (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let \mathfrak_i = R e_i, which are two-sided ideals. If each e_i is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to ''R''. An important application of an infinite direct product is the construction of a of rings (see below). Another application is a of a family of rings (cf. ).


Polynomial ring

Given a symbol ''t'' (called a variable) and a commutative ring ''R'', the set of polynomials : R = \left\ forms a commutative ring with the usual addition and multiplication, containing ''R'' as a subring. It is called the over ''R''. More generally, the set R\left _1, \ldots, t_n\right/math> of all polynomials in variables t_1, \ldots, t_n forms a commutative ring, containing R\left _i\right/math> as subrings. If ''R'' is an , then R /math> is also an integral domain; its field of fractions is the field of s. If ''R'' is a Noetherian ring, then R /math> is a Noetherian ring. If ''R'' is a unique factorization domain, then R /math> is a unique factorization domain. Finally, ''R'' is a field if and only if R /math> is a principal ideal domain. Let R \subseteq S be commutative rings. Given an element ''x'' of ''S'', one can consider the ring homomorphism : R \to S, \quad f \mapsto f(x) (that is, the ). If and , then . Because of this, the polynomial ''f'' is often also denoted by f(t). The image of the map f \mapsto f(x) is denoted by R /math>; it is the same thing as the subring of ''S'' generated by ''R'' and ''x''. Example: k\left ^2, t^3\right/math> denotes the image of the homomorphism :k
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
\to k \, f \mapsto f\left(t^2, t^3\right). In other words, it is the subalgebra of k /math> generated by ''t''2 and ''t''3. Example: let ''f'' be a polynomial in one variable, that is, an element in a polynomial ring ''R''. Then f(x+h) is an element in R /math> and f(x + h) - f(x) is divisible by ''h'' in that ring. The result of substituting zero to ''h'' in (f(x + h) - f(x))/h is f'(x), the derivative of ''f'' at ''x''. The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism \phi: R \to S and an element ''x'' in ''S'' there exists a unique ring homomorphism \overline: R \to S such that \overline(t) = x and \overline restricts to \phi. For example, choosing a basis, a satisfies the universal property and so is a polynomial ring. To give an example, let ''S'' be the ring of all functions from ''R'' to itself; the addition and the multiplication are those of functions. Let ''x'' be the identity function. Each ''r'' in ''R'' defines a constant function, giving rise to the homomorphism R \to S. The universal property says that this map extends uniquely to :R \to S, \quad f \mapsto \overline (''t'' maps to ''x'') where \overline is the defined by ''f''. The resulting map is injective if and only if ''R'' is infinite. Given a non-constant monic polynomial ''f'' in R /math>, there exists a ring ''S'' containing ''R'' such that ''f'' is a product of linear factors in S /math>. Let ''k'' be an algebraically closed field. The (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in k\left _1, \ldots, t_n\right/math> and the set of closed subvarieties of k^n. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. .) There are some other related constructions. A R ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math> consists of formal power series : \sum_0^\infty a_i t^i, \quad a_i \in R together with multiplication and addition that mimic those for convergent series. It contains R /math> as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is (in fact, ).


Matrix ring and endomorphism ring

Let ''R'' be a ring (not necessarily commutative). The set of all square matrices of size ''n'' with entries in ''R'' forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the and is denoted by M''n''(''R''). Given a right ''R''-module U, the set of all ''R''-linear maps from ''U'' to itself forms a ring with addition that is of function and multiplication that is of ; it is called the endomorphism ring of ''U'' and is denoted by \operatorname_R(U). As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: \operatorname_R(R^n) \simeq \operatorname_n(R). This is a special case of the following fact: If f: \oplus_1^n U \to \oplus_1^n U is an ''R''-linear map, then ''f'' may be written as a matrix with entries f_ in S = \operatorname_R(U), resulting in the ring isomorphism: :\operatorname_R(\oplus_1^n U) \to \operatorname_n(S), \quad f \mapsto (f_). Any ring homomorphism induces . says that if ''U'' is a simple right ''R''-module, then \operatorname_R(U) is a division ring. If \textstyle U = \bigoplus_^r U_i^ is a direct sum of ''m''''i''-copies of simple ''R''-modules U_i, then :\operatorname_R(U) \simeq \prod_^r \operatorname_ (\operatorname_R(U_i)). The states any (cf. below) is of this form. A ring ''R'' and the matrix ring M''n''(''R'') over it are : the of right modules of ''R'' is equivalent to the category of right modules over M''n''(''R''). In particular, two-sided ideals in ''R'' correspond in one-to-one to two-sided ideals in M''n''(''R'').


Limits and colimits of rings

Let ''R''''i'' be a sequence of rings such that ''R''''i'' is a subring of ''R''''i''+1 for all ''i''. Then the union (or ) of ''R''''i'' is the ring \varinjlim R_i defined as follows: it is the disjoint union of all ''R''''i'''s modulo the equivalence relation x \sim y if and only if x = y in ''R''''i'' for sufficiently large ''i''. Examples of colimits: * A polynomial ring in infinitely many variables: R[t_1, t_2, \cdots] = \varinjlim R[t_1, t_2, \cdots, t_m]. * The of s of the same characteristic \overline_p = \varinjlim \mathbf_. * The field of over a field ''k'': k(\!(t)\!) = \varinjlim t^k ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math> (it is the field of fractions of the k ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math>.) * The over a field ''k'' is \varinjlim k[U] where the limit runs over all the coordinate rings k[U] of nonempty open subsets ''U'' (more succinctly it is the of the structure sheaf at the .) Any commutative ring is the colimit of . A (or a ) of rings is defined as follows. Suppose we're given a family of rings R_i, ''i'' running over positive integers, say, and ring homomorphisms R_j \to R_i, j \ge i such that R_i \to R_i are all the identities and R_k \to R_j \to R_i is R_k \to R_i whenever k \ge j \ge i. Then \varprojlim R_i is the subring of \textstyle \prod R_i consisting of (x_n) such that x_j maps to x_i under R_j \to R_i, j \ge i. For an example of a projective limit, see .


Localization

The generalizes the construction of the of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring ''R'' and a subset ''S'' of ''R'', there exists a ring R ^/math> together with the ring homomorphism R \to R\left ^\right/math> that "inverts" ''S''; that is, the homomorphism maps elements in ''S'' to unit elements in R\left ^\right/math>, and, moreover, any ring homomorphism from ''R'' that "inverts" ''S'' uniquely factors through R\left ^\right/math>. The ring R\left ^\right/math> is called the localization of ''R'' with respect to ''S''. For example, if ''R'' is a commutative ring and ''f'' an element in ''R'', then the localization R\left[f^\right] consists of elements of the form r/f^n, \, r \in R , \, n \ge 0 (to be precise, R\left[f^\right] = R[t]/(tf - 1).) The localization is frequently applied to a commutative ring ''R'' with respect to the complement of a prime ideal (or a union of prime ideals) in ''R''. In that case S = R - \mathfrak, one often writes R_\mathfrak for R\left ^\right/math>. R_\mathfrak is then a with the \mathfrak R_\mathfrak. This is the reason for the terminology "localization". The field of fractions of an integral domain ''R'' is the localization of ''R'' at the prime ideal zero. If \mathfrak is a prime ideal of a commutative ring ''R'', then the field of fractions of R/\mathfrak is the same as the residue field of the local ring R_\mathfrak and is denoted by k(\mathfrak). If ''M'' is a left ''R''-module, then the localization of ''M'' with respect to ''S'' is given by a M\left[S^\right] = R\left[S^\right] \otimes_R M. The most important properties of localization are the following: when ''R'' is a commutative ring and ''S'' a multiplicatively closed subset * \mathfrak \mapsto \mathfrak\left ^\right/math> is a bijection between the set of all prime ideals in ''R'' disjoint from ''S'' and the set of all prime ideals in R\left ^\right/math>. * R\left[S^\right] = \varinjlim R\left[f^\right], ''f'' running over elements in ''S'' with partial ordering given by divisibility. * The localization is exact: *: 0 \to M'\left[S^\right] \to M\left[S^\right] \to M''\left[S^\right] \to 0 is exact over R\left ^\right/math> whenever 0 \to M' \to M \to M'' \to 0 is exact over ''R''. * Conversely, if 0 \to M'_\mathfrak \to M_\mathfrak \to M''_\mathfrak \to 0 is exact for any maximal ideal \mathfrak, then 0 \to M' \to M \to M'' \to 0 is exact. * A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.) In , a amounts to making some morphisms isomorphisms. An element in a commutative ring ''R'' may be thought of as an endomorphism of any ''R''-module. Thus, categorically, a localization of ''R'' with respect to a subset ''S'' of ''R'' is a from the category of ''R''-modules to itself that sends elements of ''S'' viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, ''R'' then maps to R\left ^\right/math> and ''R''-modules map to R\left ^\right/math>-modules.)


Completion

Let ''R'' be a commutative ring, and let ''I'' be an ideal of ''R''. The of ''R'' at ''I'' is the projective limit \hat = \varprojlim R/I^n; it is a commutative ring. The canonical homomorphisms from ''R'' to the quotients R/I^n induce a homomorphism R \to \hat. The latter homomorphism is injective if ''R'' is a Noetherian integral domain and ''I'' is a proper ideal, or if ''R'' is a Noetherian local ring with maximal ideal ''I'', by . The construction is especially useful when ''I'' is a maximal ideal. The basic example is the completion of Z at the principal ideal (''p'') generated by a prime number ''p''; it is called the ring of and is denoted Z''p''. The completion can in this case be constructed also from the on Q. The ''p''-adic absolute value on Q is a map x \mapsto , x, from Q to R given by , n, _p=p^ where v_p(n) denotes the exponent of ''p'' in the prime factorization of a nonzero integer ''n'' into prime numbers (we also put , 0, _p=0 and , m/n, _p = , m, _p/, n, _p). It defines a distance function on Q and the completion of Q as a is denoted by Q''p''. It is again a field since the field operations extend to the completion. The subring of Q''p'' consisting of elements ''x'' with , x, _p \le 1 is isomorphic to Z''p''. Similarly, the formal power series ring R[] is the completion of R /math> at (t) (see also ) A complete ring has much simpler structure than a commutative ring. This owns to the , which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of .


Rings with generators and relations

The most general way to construct a ring is by specifying generators and relations. Let ''F'' be a (that is, free algebra over the integers) with the set ''X'' of symbols, that is, ''F'' consists of polynomials with integral coefficients in noncommuting variables that are elements of ''X''. A free ring satisfies the universal property: any function from the set ''X'' to a ring ''R'' factors through ''F'' so that F \to R is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring. Now, we can impose relations among symbols in ''X'' by taking a quotient. Explicitly, if ''E'' is a subset of ''F'', then the quotient ring of ''F'' by the ideal generated by ''E'' is called the ring with generators ''X'' and relations ''E''. If we used a ring, say, ''A'' as a base ring instead of Z, then the resulting ring will be over ''A''. For example, if E = \, then the resulting ring will be the usual polynomial ring with coefficients in ''A'' in variables that are elements of ''X'' (It is also the same thing as the over ''A'' with symbols ''X''.) In the category-theoretic terms, the formation S \mapsto \text S is the left adjoint functor of the from the to Set (and it is often called the free ring functor.) Let ''A'', ''B'' be algebras over a commutative ring ''R''. Then the tensor product of ''R''-modules A \otimes_R B is an ''R''-algebra with multiplication characterized by (x \otimes u) (y \otimes v) = xy \otimes uv. See also: ', '.


Special kinds of rings


Domains

A ring with no nonzero s is called a . A commutative domain is called an . The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a (UFD), an integral domain in which every nonunit element is a product of s (an element is prime if it generates a .) The fundamental question in is on the extent to which the in a , where an "ideal" admits prime factorization, fails to be a PID. Among theorems concerning a PID, the most important one is the . The theorem may be illustrated by the following application to linear algebra. Let ''V'' be a finite-dimensional vector space over a field ''k'' and f: V \to V a linear map with minimal polynomial ''q''. Then, since k /math> is a unique factorization domain, ''q'' factors into powers of distinct irreducible polynomials (that is, prime elements): :q = p_1^ \ldots p_s^. Letting t \cdot v = f(v), we make ''V'' a ''k''[''t'']-module. The structure theorem then says ''V'' is a direct sum of s, each of which is isomorphic to the module of the form k[t]/\left(p_i^\right). Now, if p_i(t) = t - \lambda_i, then such a cyclic module (for p_i) has a basis in which the restriction of ''f'' is represented by a . Thus, if, say, ''k'' is algebraically closed, then all p_i's are of the form t - \lambda_i and the above decomposition corresponds to the of ''f''. In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a . A regular local ring is a UFD. The following is a chain of that describes the relationship between rings, domains and fields: :


Division ring

A is a ring such that every non-zero element is a unit. A commutative division ring is a . A prominent example of a division ring that is not a field is the ring of s. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every ''finite'' domain (in particular finite division ring) is a field; in particular commutative (the ). Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the . A , introduced by , is a generalization of a .


Semisimple rings

A ' is a direct sum of simple modules. A ' is a ring that is semisimple as a left module (or right module) over itself.


Examples

* A is semisimple (and ). * For any division ring and positive integer , the matrix ring is semisimple (and ). * For a field and finite group , the group ring is semisimple if and only if the of does not divide the of (). * s are semisimple. The over a field is a , but it is not semisimple. The same holds for a .


Properties

Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.) For a ring , the following are equivalent: * is semisimple. * is and . * is a finite \textstyle \prod_^r \operatorname_(D_i) where each is a positive integer, and each is a division ring (). Semisimplicity is closely related to separability. A unital associative algebra ''A'' over a field ''k'' is said to be if the base extension A \otimes_k F is semisimple for every F/k. If ''A'' happens to be a field, then this is equivalent to the usual definition in field theory (cf. .)


Central simple algebra and Brauer group

For a field ''k'', a ''k''-algebra is central if its center is ''k'' and is simple if it is a . Since the center of a simple ''k''-algebra is a field, any simple ''k''-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a ''k''-algebra. The matrix ring of size ''n'' over a ring ''R'' will be denoted by R_n. The states any automorphism of a central simple algebra is inner. Two central simple algebras ''A'' and ''B'' are said to be ''similar'' if there are integers ''n'' and ''m'' such that A \otimes_k k_n \approx B \otimes_k k_m. Since k_n \otimes_k k_m \simeq k_, the similarity is an equivalence relation. The similarity classes [A] with the multiplication [A][B] = \left[A \otimes_k B\right] form an abelian group called the of ''k'' and is denoted by \operatorname(k). By the , a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. For example, \operatorname(k) is trivial if ''k'' is a finite field or an algebraically closed field (more generally ; cf. ). \operatorname(\mathbf) has order 2 (a special case of the ). Finally, if ''k'' is a nonarchimedean (for example, \mathbf_p), then \operatorname(k) = \mathbf/\mathbf through the . Now, if ''F'' is a field extension of ''k'', then the base extension - \otimes_k F induces \operatorname(k) \to \operatorname(F). Its kernel is denoted by \operatorname(F/k). It consists of [A] such that A \otimes_k F is a matrix ring over ''F'' (that is, ''A'' is split by ''F''.) If the extension is finite and Galois, then \operatorname(F/k) is canonically isomorphic to H^2\left(\operatorname(F/k), k^*\right). s generalize the notion of central simple algebras to a commutative local ring.


Valuation ring

If ''K'' is a field, a ''v'' is a group homomorphism from the multiplicative group ''K'' to a totally ordered abelian group ''G'' such that, for any ''f'', ''g'' in ''K'' with ''f'' + ''g'' nonzero, The of ''v'' is the subring of ''K'' consisting of zero and all nonzero ''f'' such that . Examples: See also: and .


Rings with extra structure

A ring may be viewed as an (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example: * An is a ring that is also a over a field ''K'' such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of ''n''-by-''n'' matrices over the real field R has dimension ''n''2 as a real vector space. * A ring ''R'' is a if its set of elements ''R'' is given a which makes the addition map ( + : R\times R \to R\,) and the multiplication map ( \cdot : R\times R \to R\,) to be both as maps between topological spaces (where ''X'' × ''X'' inherits the or any other product in the category). For example, ''n''-by-''n'' matrices over the real numbers could be given either the , or the , and in either case one would obtain a topological ring. * A is a commutative ring ''R'' together with operations that are like ''n''-th s: ::\lambda^n(x + y) = \sum_0^n \lambda^i(x) \lambda^(y). :For example, Z is a λ-ring with \lambda^n(x) = \binom, the s. The notion plays a central rule in the algebraic approach to the . * A is a ring with a that is compatible with ring operations.


Some examples of the ubiquity of rings

Many different kinds of s can be fruitfully analyzed in terms of some .


Cohomology ring of a topological space

To any ''X'' one can associate its integral :H^*(X,\mathbf) = \bigoplus_^ H^i(X,\mathbf), a . There are also s H_i(X,\mathbf) of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the s and , for which the methods of are not well-suited. s were later defined in terms of homology groups in a way which is roughly analogous to the dual of a . To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the . However, the advantage of the cohomology groups is that there is a , which is analogous to the observation that one can multiply pointwise a ''k''- and an ''l''-multilinear form to get a ()-multilinear form. The ring structure in cohomology provides the foundation for es of s, intersection theory on manifolds and , and much more.


Burnside ring of a group

To any is associated its which uses a ring to describe the various ways the group can on a finite set. The Burnside ring's additive group is the whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the : the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.


Representation ring of a group ring

To any or is associated its or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from , which is more or less the given a ring structure.


Function field of an irreducible algebraic variety

To any irreducible is associated its . The points of an algebraic variety correspond to s contained in the function field and containing the . The study of makes heavy use of to study geometric concepts in terms of ring-theoretic properties. studies maps between the subrings of the function field.


Face ring of a simplicial complex

Every has an associated face ring, also called its . This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in . In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of s.


Category-theoretic description

Every ring can be thought of as a in Ab, the (thought of as a under the ). The monoid action of a ring ''R'' on an abelian group is simply an . Essentially, an ''R''-module is a generalization of the notion of a – where rather than a vector space over a field, one has a "vector space over a ring". Let be an abelian group and let End(''A'') be its (see above). Note that, essentially, End(''A'') is the set of all morphisms of ''A'', where if ''f'' is in End(''A''), and ''g'' is in End(''A''), the following rules may be used to compute and : * (''f'' + ''g'')(''x'') = ''f''(''x'') + ''g''(''x'') * (''f'' ⋅ ''g'')(''x'') = ''f''(''g''(''x'')), where + as in is addition in ''A'', and function composition is denoted from right to left. Therefore, to any abelian group, is a ring. Conversely, given any ring, , is an abelian group. Furthermore, for every ''r'' in ''R'', right (or left) multiplication by ''r'' gives rise to a morphism of , by right (or left) distributivity. Let . Consider those s of ''A'', that "factor through" right (or left) multiplication of ''R''. In other words, let End''R''(''A'') be the set of all morphisms ''m'' of ''A'', having the property that . It was seen that every ''r'' in ''R'' gives rise to a morphism of ''A'': right multiplication by ''r''. It is in fact true that this association of any element of ''R'', to a morphism of ''A'', as a function from ''R'' to End''R''(''A''), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian ''X''-group (by ''X''-group, it is meant a group with ''X'' being its ). In essence, the most general form of a ring, is the endomorphism group of some abelian ''X''-group. Any ring can be seen as a with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. s between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of s closed under addition and under composition with arbitrary morphisms.


Generalization

Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.


Rng

A is the same as a ring, except that the existence of a multiplicative identity is not assumed.


Nonassociative ring

A is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a . There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.


Semiring

A (sometimes ''rig'') is obtained by weakening the assumption that (''R'', +) is an abelian group to the assumption that (''R'', +) is a commutative monoid, and adding the axiom that for all ''a'' in ''R'' (since it no longer follows from the other axioms). Examples: * the non-negative integers \ with ordinary addition and multiplication; * the .


Other ring-like objects


Ring object in a category

Let ''C'' be a category with finite . Let pt denote a of ''C'' (an empty product). A ring object in ''C'' is an object ''R'' equipped with morphisms R \times R\;\stackrel\to\,R (addition), R \times R\;\stackrel\to\,R (multiplication), \operatorname\stackrel\to\,R (additive identity), R\;\stackrel\to\,R (additive inverse), and \operatorname\stackrel\to\,R (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object ''R'' equipped with a factorization of its functor of points h_R = \operatorname(-,R) : C^ \to \mathbf through the category of rings: C^ \to \mathbf \stackrel\longrightarrow \mathbf.


Ring scheme

In algebraic geometry, a ring scheme over a base is a ring object in the category of -schemes. One example is the ring scheme over , which for any commutative ring returns the ring of -isotypic Witt vectors of length over .Serre, p. 44.


Ring spectrum

In , a is a ''X'' together with a multiplication \mu \colon X \wedge X \to X and a unit map S \to X from the ''S'', such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a in a good category of spectra such as the category of .


See also

* * * * * * * * * Special types of rings: * * * * * * * * and s * * * * * * * and


Notes


Citations


References


General references

* * * * * . * * * * * * * * * . * * * * . * * * . * . * * *


Special references

* * * * * * * * * * * * * * * * * * * *


Primary sources

* * *


Historical references


History of ring theory at the MacTutor Archive
* and (1996) ''A Survey of Modern Algebra'', 5th ed. New York: Macmillan. * Bronshtein, I. N. and Semendyayev, K. A. (2004) , 4th ed. New York: Springer-Verlag . * Faith, Carl (1999) ''Rings and things and a fine array of twentieth century associative algebra''. Mathematical Surveys and Monographs, 65. . * Itô, K. editor (1986) "Rings." §368 in ''Encyclopedic Dictionary of Mathematics'', 2nd ed., Vol. 2. Cambridge, MA: . * (1996) "The Genesis of the Abstract Ring Concept", 103: 417–424 * Kleiner, I. (1998) "From numbers to rings: the early history of ring theory", 53: 18–35. * (1985) ''A History of Algebra'', Springer-Verlag, {{DEFAULTSORT:Ring (Mathematics) Ring theory