mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a subring of a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

is a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of that is itself a ring when

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

s of addition and multiplication on ''R'' are restricted to the subset, and that shares the same

multiplicative identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

as .In general, not all subsets of a ring are rings.

Definition

A subring of a ring is a subset of that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of and a

submonoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

of . Equivalently, is a subring

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

it contains the multiplicative identity of , and is closed under multiplication and subtraction. This is sometimes known as the ''subring test''.

Variations

Some mathematicians define rings without requiring the existence of a multiplicative identity (see '). In this case, a subring of is a subset of that is a ring for the operations of (this does imply it contains the additive identity of ). This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all ideals become subrings, and they may have a multiplicative identity that differs from the one of . With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of that is a subring of is itself.

Examples

* The

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...

\Z

is a subring of both the field of

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s and the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

\Z /math>. * \mathbb and its quotients \mathbb/n\mathbb have no subrings (with multiplicative identity) other than the full ring. * Every ring has a unique smallest subring, isomorphic to some ring \mathbb/n\mathbb with ''n'' a nonnegative integer (see '' Characteristic''). The integers \mathbb correspond to  in this statement, since \mathbb is isomorphic to \mathbb/0\mathbb .

* The

center of a ring In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(''R''); 'Z' stands for the German word ''Zentrum'', meaning "center". ...

is a subring of , and is an

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

over its center.

Subring generated by a set

A special kind of subring of a ring is the subring generated by a subset , which is defined as the intersection of all subrings of containing . The subring generated by is also the set of all

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

s with integer coefficients of products of elements of , including the additive identity ("empty combination") and multiplicative identity ("empty product"). Any intersection of subrings of is itself a subring of ; therefore, the subring generated by (denoted here as ) is indeed a subring of . This subring is the smallest subring of containing ; that is, if is any other subring of containing , then . Since itself is a subring of , if is generated by , it is said that the ring is ''generated by'' .

Ring extension

Subrings generalize some aspects of

field extensions Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

. If is a subring of a ring , then equivalently is said to be a ring extensionNot to be confused with the ring-theoretic analog of a

group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...

. of .

Adjoining

If is a ring and is a subring of generated by , where is a subring, then is a ring extension and is said to be ''adjoined to'' , denoted . Individual elements can also be adjoined to a subring, denoted . For example, the ring of

Gaussian integers In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...

\Z /math> is a subring of \C generated by \Z \cup \, and thus is the adjunction of the

imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

\Z

Prime subring

The intersection of all subrings of a ring is a subring that may be called the ''prime subring'' of by analogy with

prime field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is wid ...

s. The prime subring of a ring is a subring of the center of , which is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

either to the ring

\Z

of the

integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

or to the ring of the integers modulo , where is the smallest positive integer such that the sum of copies of equals .

Notes

References

General references

* * {{cite book , last1=Sharpe , first1=David , title=Rings and factorization , url=https://archive.org/details/ringsfactorizati0000shar , url-access=registration , publisher=

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, date=1987 , isbn=0-521-33718-6 , page
15–17
} Ring theory