Ring homomorphism

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][a]}

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,^[b] 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
  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}.
• The
  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
  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.
• As the initial object is not isomorphic to the terminal object, there is no
  zero object
  in the category of rings; in particular, the zero ring is not a zero object in the category of rings.

• 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
  .

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 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 r₁ and r₂ to the same element of S. Consider the two maps g₁ and g₂ from Z[x] to R that map x to r₁ and r₂, respectively; f ∘ g₁ and f ∘ g₂ are identical, but since f is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from

See also

• Change of rings

Notes

Citations

References