TheInfoList

In
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, a branch of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a ring homomorphism is a structure-preserving
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
between two
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserving: ::$f\left(a+b\right)=f\left(a\right)+f\left(b\right)$ for all ''a'' and ''b'' in ''R'', :multiplication preserving: ::$f\left(ab\right)=f\left(a\right)f\left(b\right)$ for all ''a'' and ''b'' in ''R'', :and unit (multiplicative identity) preserving: ::$f\left(1_R\right)=1_S$. Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. If in addition ''f'' is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, then its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
''f''−1 is also a ring homomorphism. In this case, ''f'' is called a ring isomorphism, and the rings ''R'' and ''S'' are called ''isomorphic''. From the standpoint of ring theory, isomorphic rings cannot be distinguished. If ''R'' and ''S'' are rngs, then the corresponding notion is that of a rng homomorphism, defined as above except without the third condition ''f''(1''R'') = 1''S''. A rng homomorphism between (unital) rings need not be a ring homomorphism. The
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
of two ring homomorphisms is a ring homomorphism. It follows that the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of all rings forms a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
with ring homomorphisms as the
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s (cf. the
category of rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

# Properties

Let $f \colon R \rightarrow S$ be a ring homomorphism. Then, directly from these definitions, one can deduce: * ''f''(0''R'') = 0''S''. * ''f''(−''a'') = −''f''(''a'') for all ''a'' in ''R''. * For any
unit element In the branch of abstract algebra known as ring theory, a unit of a ring (mathematics), ring R is any element u \in R that has a multiplicative inverse in R: an element v \in R such that :vu = uv = 1, where is the multiplicative identity. The s ...
''a'' in ''R'', ''f''(''a'') is a unit element such that . In particular, ''f'' induces a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from the (multiplicative) group of units of ''R'' to the (multiplicative) group of units of ''S'' (or of im(''f'')). * The
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of ''f'', denoted im(''f''), is a subring of ''S''. * The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of ''f'', defined as , is an
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
in ''R''. Every ideal in a ring ''R'' arises from some ring homomorphism in this way. * The homomorphism ''f'' is injective if and only if . * If there exists a ring homomorphism then the
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
of ''S''
divides In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
the characteristic of ''R''. This can sometimes be used to show that between certain rings ''R'' and ''S'', no ring homomorphisms can exist. * If ''Rp'' is the smallest
subring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
contained in ''R'' and ''Sp'' is the smallest subring contained in ''S'', then every ring homomorphism induces a ring homomorphism . * If ''R'' is a
field 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 grassl ...
(or more generally a
skew-fieldIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
) and ''S'' is not the
zero ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
, then ''f'' is injective. * If both ''R'' and ''S'' are
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
, then im(''f'') is a subfield of ''S'', so ''S'' can be viewed as a
field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of ''R''. *If ''R'' and ''S'' are commutative and I is an ideal of ''S'' then ''f''−1(I) is an ideal of ''R''. * If ''R'' and ''S'' are commutative and ''P'' is a
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
of ''S'' then ''f''−1(''P'') is a prime ideal of ''R''. *If ''R'' and ''S'' are commutative, M is a
maximal ideal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of ''S'', and ''f'' is surjective, then ''f''−1(M) is a maximal ideal of ''R''. * If ''R'' and ''S'' are commutative and ''S'' is an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, then ker(''f'') is a prime ideal of ''R''. * If ''R'' and ''S'' are commutative, ''S'' is a field, and ''f'' is surjective, then ker(''f'') is a
maximal ideal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of ''R''. * If ''f'' is surjective, ''P'' is prime (maximal) ideal in ''R'' and , then ''f''(''P'') is prime (maximal) ideal in ''S''. Moreover, *The composition of ring homomorphisms is a ring homomorphism. *For each ring ''R'', the identity map is a ring homomorphism. *Therefore, the class of all rings together with ring homomorphisms forms a category, the
category of rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
. *The zero map sending every element of ''R'' to 0 is a ring homomorphism only if ''S'' is the
zero ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
(the ring whose only element is zero). * For every ring ''R'', there is a unique ring homomorphism . This says that the ring of integers is an
initial object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
of rings. * For every ring ''R'', there is a unique ring homomorphism from ''R'' to the zero ring. This says that the zero ring is a
terminal object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
in the category of rings.

# Examples

* The function , defined by is a
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
ring homomorphism with kernel ''n''Z (see
modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
). * The function defined by is a rng homomorphism (and rng endomorphism), with kernel 3Z6 and image 2Z6 (which is isomorphic to Z3). * There is no ring homomorphism for . * The
complex conjugation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is a ring homomorphism (this is an example of a ring automorphism.) * If ''R'' and ''S'' are rings, the zero function from ''R'' to ''S'' is a ring homomorphism if and only if ''S'' is the
zero ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
. (Otherwise it fails to map 1''R'' to 1''S''.) On the other hand, the zero function is always a rng homomorphism. * If R 'X''denotes the ring of all
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s in the variable ''X'' with coefficients in the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s R, and C denotes the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, then the function defined by (substitute the imaginary unit ''i'' for the variable ''X'' in the polynomial ''p'') is a surjective ring homomorphism. The kernel of ''f'' consists of all polynomials in R 'X''which are divisible by . * If is a ring homomorphism between the rings ''R'' and ''S'', then ''f'' induces a ring homomorphism between the
matrix ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s . * A unital
algebra homomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
between unital
associative algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s over a commutative ring ''R'' is a ring homomorphism that is also ''R''-linear.

# Non-examples

*Given a product of rings $S = R_1 \times R_2$, the natural inclusion $R_1 \to S, x \mapsto \left(x, 0\right)$ is not a ring homomorphism (unless $R_2$ is the zero ring); this is because the map does not send the multiplicative identity of $R_1$ to that of $S$, namely $\left(1, 1\right)$.

# The category of rings

## Endomorphisms, isomorphisms, and automorphisms

* A ring endomorphism is a ring homomorphism from a ring to itself. * A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
as a function on the underlying sets. If there exists a ring isomorphism between two rings ''R'' and ''S'', then ''R'' and ''S'' are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven rngs of order 4. * A ring automorphism is a ring isomorphism from a ring to itself.

## Monomorphisms and epimorphisms

Injective ring homomorphisms are identical to
monomorphism In the context of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...
s in the category of rings: If is a monomorphism that is not injective, then it sends some ''r''1 and ''r''2 to the same element of ''S''. Consider the two maps ''g''1 and ''g''2 from Z 'x''to ''R'' that map ''x'' to ''r''1 and ''r''2, respectively; and are identical, but since ''f'' is a monomorphism this is impossible. However, surjective ring homomorphisms are vastly different from
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
s in the category of rings. For example, the inclusion is a ring epimorphism, but not a surjection. However, they are exactly the same as the
strong epimorphism Image:Epimorphism scenarios.svg, 220px In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is Cancellation property, right-cancellative in the sense that, for all obj ...
s.