ring homomorphism

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In
ring theory In algebra, ring theory is the study of 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 structure of rings, their re ...
, a branch of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, a ring homomorphism is a structure-preserving
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
between two
rings Ring 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 :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
. 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, then its inverse ''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 and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
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 differentl ...
of all rings forms a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
with ring homomorphisms as the morphisms (cf. the
category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of ring ...
). 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 algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for thi ...
''a'' in ''R'', ''f''(''a'') is a unit element such that . In particular, ''f'' induces a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
from the (multiplicative) group of units of ''R'' to the (multiplicative) group of units of ''S'' (or of im(''f'')). * The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 learn ...
of ''f'', defined as , is an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
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 of ''S''
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
the characteristic of ''R''. This can sometimes be used to show that between certain rings ''R'' and ''S'', no ring homomorphisms exists. * If ''Rp'' is the smallest subring 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 grass ...
(or more generally a
skew-field In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
) and ''S'' is not the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for a ...
, then ''f'' is injective. * If both ''R'' and ''S'' are
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
, then im(''f'') is a subfield of ''S'', so ''S'' can be viewed as a field extension 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, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
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, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
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, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
, 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, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
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 whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of ring ...
. *The zero map sending every element of ''R'' to 0 is a ring homomorphism only if ''S'' is the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for a ...
(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, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
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, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
in the category of rings.

# Examples

* The function , defined by is a surjective ring homomorphism with kernel ''n''Z (see
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
). * The complex conjugation is a ring homomorphism (this is an example of a ring automorphism). * For a ring ''R'' of prime characteristic ''p'', is a ring endomorphism called the
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
. * 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, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for a ...
. (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, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s in the variable ''X'' with coefficients in the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s R, and C denotes the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
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''that 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, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
s . *Let ''V'' be a vector space over a field ''k''. Then the map $\rho : k \to \operatorname\left(V\right)$ given by $\rho\left(a\right)v = av$ is a ring homomorphism. More generally, given an abelian group ''M'', a module structure on ''M'' over a ring ''R'' is equivalent to giving a ring homomorphism $R \to \operatorname\left(M\right)$. * A unital
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF ...
between unital
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
s over a commutative ring ''R'' is a ring homomorphism that is also ''R''-linear.

# Non-examples

* The function defined by is a rng homomorphism (and rng endomorphism), with kernel 3Z/6Z and image 2Z/6Z (which is isomorphic to Z/3Z). * There is no ring homomorphism for any . * If ''R'' and ''S'' are rings, the inclusion $R \to R \times S$ sending each ''r'' to (''r'',0) is a rng homomorphism, but not a ring homomorphism (if ''S'' is not the zero ring), since it does not map the multiplicative identity 1 of ''R'' to the multiplicative identity (1,1) of $R \times S$.

# 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, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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 or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
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 In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
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 In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
s.

# See also

*
Change of rings In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operatorn ...
*
Ring extension In commutative algebra, a ring extension is a ring homomorphism R\to S of commutative rings, which makes an -algebra. In this article, a ring extension of a ring ''R'' by an abelian group ''I'' is a pair of a ring ''E'' and a surjective ring ho ...

# References

* * * * * * * {{refend Ring theory Morphisms