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In ring theory, a branch of abstract algebra, a **ring homomorphism** is a structure-preserving function between two rings. More explicitly, if *R* and *S* are rings, then a ring homomorphism is a function *f* : *R* → *S* such that *f* is:^{ [1] }^{ [2] }^{ [3] }^{ [4] }^{ [5] }^{ [6] }^{ [7] }^{ [lower-alpha 1] }

- Properties
- Examples
- Non-examples
- The category of rings
- Endomorphisms, isomorphisms, and automorphisms
- Monomorphisms and epimorphisms
- See also
- Citations
- Notes
- References

- addition preserving:
- for all
*a*and*b*in*R*,

- for all

- multiplication preserving:
- for all
*a*and*b*in*R*,

- for all

- and unit (multiplicative identity) preserving:
- .

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**,^{ [lower-alpha 2] } 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 of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

Let 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
*a*in*R*,*f*(*a*) is a unit element such that*f*(*a*^{−1}) =*f*(*a*)^{−1}. In particular,*f*induces a group homomorphism from the (multiplicative) group of units of*R*to the (multiplicative) group of units of*S*(or of im(*f*)). - The image of
*f*, denoted im(*f*), is a subring of*S*. - The kernel of
*f*, defined as ker(*f*) = {*a*in*R*:*f*(*a*) = 0_{S}}, is an ideal 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 ker(*f*) = {0_{R}}. - If there exists a ring homomorphism
*f*:*R*→*S*then the characteristic of*S*divides the characteristic of*R*. This can sometimes be used to show that between certain rings*R*and*S*, no ring homomorphisms*R*→*S*exists. - If
*R*is the smallest subring contained in_{p}*R*and*S*is the smallest subring contained in_{p}*S*, then every ring homomorphism*f*:*R*→*S*induces a ring homomorphism*f*:_{p}*R*→_{p}*S*._{p} - If
*R*is a field (or more generally a skew-field) and*S*is not the zero ring, then*f*is injective. - If both
*R*and*S*are fields, 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 of*S*then*f*^{−1}(*P*) is a prime ideal of*R*. - If
*R*and*S*are commutative,*M*is a maximal ideal 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, 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 of*R*. - If
*f*is surjective,*P*is prime (maximal) ideal in*R*and ker(*f*) ⊆*P*, 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*R*→*R*is a ring homomorphism. - Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
- The zero map
*R*→*S*sending every element of*R*to 0 is a ring homomorphism only if*S*is the zero ring (the ring whose only element is zero). - For every ring
*R*, there is a unique ring homomorphism**Z**→*R*. This says that the ring of integers is an initial object in the category 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 the category of rings.

- The function
*f*:**Z**→**Z**/*n***Z**, defined by*f*(*a*) = [*a*]_{n}=*a*mod*n*is a surjective ring homomorphism with kernel*n***Z**(see modular arithmetic). - The complex conjugation
**C**→**C**is a ring homomorphism (this is an example of a ring automorphism). - For a ring
*R*of prime characteristic*p*,**R**→**R**,*x*→*x*^{p}is a ring endomorphism called the Frobenius endomorphism. - 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. (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 polynomials in the variable*X*with coefficients in the real numbers**R**, and**C**denotes the complex numbers, then the function*f*:**R**[*X*] →**C**defined by*f*(*p*) =*p*(*i*) (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*X*^{2}+ 1. - If
*f*:*R*→*S*is a ring homomorphism between the rings*R*and*S*, then*f*induces a ring homomorphism between the matrix rings M_{n}(*R*) → M_{n}(*S*). - Let
*V*be a vector space over a field*k*. Then the map given by 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 . - A unital algebra homomorphism between unital associative algebras over a commutative ring
*R*is a ring homomorphism that is also*R*-linear.

- The function
*f*:**Z**/6**Z**→**Z**/6**Z**defined by*f*([*a*]_{6}) = [4*a*]_{6}is a rng homomorphism (and rng endomorphism), with kernel 3**Z**/6**Z**and image 2**Z**/6**Z**(which is isomorphic to**Z**/3**Z**). - There is no ring homomorphism
**Z**/*n***Z**→**Z**for any*n*≥ 1. - If
*R*and*S*are rings, the inclusion 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 .

- 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 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.

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If *f* : *R* → *S* 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; *f* ∘ *g*_{1} and *f* ∘ *g*_{2} are identical, but since *f* is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion **Z** ⊆ **Q** is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.

- ↑ Artin 1991, p. 353.
- ↑ Atiyah & Macdonald 1969, p. 2.
- ↑ Bourbaki 1998, p. 102.
- ↑ Eisenbud 1995, p. 12.
- ↑ Jacobson 1985, p. 103.
- ↑ Lang 2002, p. 88.
- ↑ Hazewinkel 2004, p. 3.

- ↑ Hazewinkel initially defines "ring" without the requirement of a 1, but very soon states that from now on, all rings will have a 1.
- ↑ Some authors do not require a ring to contain a multiplicative identity; instead of "rng", "ring", and "rng homomorphism", they use the terms "ring", "ring with identity", and "ring homomorphism", respectively. Because of this, some other authors, to avoid ambiguity, explicitly specify that rings are unital and that homomorphisms preserve the identity.

In mathematics, an **associative algebra***A* is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field *K*. The addition and multiplication operations together give *A* the structure of a ring; the addition and scalar multiplication operations together give *A* the structure of a vector space over *K*. In this article we will also use the term *K*-algebra to mean an associative algebra over the field *K*. A standard first example of a *K*-algebra is a ring of square matrices over a field *K*, with the usual matrix multiplication.

In mathematics, given two groups, and, a **group homomorphism** from to is a function *h* : *G* → *H* such that for all *u* and *v* in *G* it holds that

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In mathematics, the endomorphisms of an abelian group *X* form a ring. This ring is called the **endomorphism ring** of *X*, denoted by End(*X*); the set of all homomorphisms of *X* into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.

In ring theory, a branch of abstract algebra, a **quotient ring**, also known as **factor ring**, **difference ring** or **residue class ring**, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring *R* and a two-sided ideal *I* in *R*, a new ring, the quotient ring *R* / *I*, is constructed, whose elements are the cosets of *I* in *R* subject to special + and ⋅ operations.

In mathematics, an **algebra over a field** is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In commutative algebra and algebraic geometry, **localization** is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module *R*, so that it consists of fractions such that the denominator *s* belongs to a given subset *S* of *R*. If *S* is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers.

In algebra, a **unit** of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that

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In category theory, a branch of abstract mathematics, an **equivalence of categories** is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In algebra, a **module homomorphism** is a function between modules that preserves the module structures. Explicitly, if *M* and *N* are left modules over a ring *R*, then a function is called an *R*-*module homomorphism* or an *R*-*linear map* if for any *x*, *y* in *M* and *r* in *R*,

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 *n* × *n* matrices with entries in *R* is a matrix ring denoted M_{n}(*R*). Some sets of infinite matrices form **infinite matrix rings**. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs.

In ring theory, a branch of mathematics, the **zero ring** or **trivial ring** is the unique ring consisting of one element.

In mathematics, and more specifically in abstract algebra, a **rng** is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term "rng" is meant to suggest that it is a "ring" without "i", that is, without the requirement for an "identity element".

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

- Artin, Michael (1991).
*Algebra*. Englewood Cliffs, N.J.: Prentice Hall. - Atiyah, Michael F.; Macdonald, Ian G. (1969),
*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802 - Bourbaki, N. (1998).
*Algebra I, Chapters 1–3*. Springer. - Eisenbud, David (1995).
*Commutative algebra with a view toward algebraic geometry*. Graduate Texts in Mathematics. Vol. 150. New York: Springer-Verlag. xvi+785. ISBN 0-387-94268-8. MR 1322960. - Hazewinkel, Michiel (2004).
*Algebras, rings and modules*. Springer-Verlag. ISBN 1-4020-2690-0. - Jacobson, Nathan (1985).
*Basic algebra I*(2nd ed.). ISBN 9780486471891. - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

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