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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]}

addition preserving:

f(a + b) = f(a) + f(b) for all a and b in R,

multiplication preserving:

f(ab) = f(a)f(b) for all a and b in R,

and unit (multiplicative identity) preserving:

f(1_R) = 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,^{[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.

Properties

Let f : R → 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 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 homomorphism R → S exists.
• 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.
• 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 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 S → T and R → S is a ring homomorphism R → T.
• 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 that sends 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.

Examples

• The function f : Z → Z/nZ, defined by f(a) = [a]_n = a mod n is a surjective ring homomorphism with kernel nZ (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 X² + 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 ρ : k → End(V) given by ρ(a)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 → End(M).
• A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear.

Non-examples

• The function f : Z/6Z → Z/6Z defined by f([a]₆) = [4a]₆ 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 Z/nZ → Z for any n ≥ 1.
• If R and S are rings, the inclusion R → R × S that sends 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 × S.

Category of rings

See also

• Change of rings

Notes

Citations

References