In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Partial plot of a function f. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that

• f(a + b) = f(a) + f(b) for all a and b in R
• f(ab) = f(a) f(b) for all a and b in R
• f(1) = 1

(If one does not require rings to have a multiplicative identity then the last condition is dropped.)

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. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

Properties

Directly from these definitions, one can deduce:

• f(0) = 0
• f(−a) = −f(a)
• If a has a multiplicative inverse in R, then f(a) has a multiplicative inverse in S and we have f(a⁻¹) = (f(a))⁻¹. Therefore, f induces a group homomorphism from the group of units of R to the group of units of S.
• The kernel of f, defined as ker(f) = {a ∈ R : f(a) = 0} is an ideal in R. Every ideal in a commutative ring R arises from some ring homomorphism in this way, but this is never true for a non-commutative ring. f is injective if and only if the ker(f) = {0}. Note that in general, for rings with identity the kernel of a ring homomorphism is not a subring since it will not contain the multiplicative identity.
• The image of f, im(f), is a subring of S.
• If f is bijective, then its inverse f⁻¹ is also a ring homomorphism. f is called an isomorphism in this case, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
• If R_p is the smallest subring contained in R and S_p is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism f_p : R_p → S_p. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms R → S can exist.
• If R is a field, then f is either injective or f is the zero function. (Note, however, that if f preserves the multiplicative identity, then it cannot be the zero function.)
• If both R and S are fields, then im(f) is a subfield of S (if f is not the zero function).
• If R and S are commutative and S has no zero divisors, then ker(f) is a prime ideal of R.
• 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.

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) · h(v) From this property, one can deduce that h maps the identity element... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...

Examples

• The function f : Z → Z_n, defined by f(a) = [a]_n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
• There is no ring homomorphism Z_n → Z for n > 1.
• 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] which are divisible by X² + 1.
• If f : R → S is a ring homomorphism between the commutative rings R and S, then f induces a ring homomorphism between the matrix rings M_n(R) → M_n(S).

In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ...

Types of ring homomorphisms

• A bijective ring homomorphism is called ring isomorphism.
• A ring homomorphism whose domain is the same as its range is called a ring endomorphism.

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f:R→S is a monomorphism which is not injective, then it sends some r₁ and r₂ to the same element of S. Consider the two maps g₁ and g₂ from Z[x] to R which map x to r₁ and r₂, respectively; f o g₁ and f o g₂ are identical, but since f is a monomorphism this is impossible. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...

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. In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ...

See also

• homomorphism
• ring theory