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]

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,

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. On the other hand, neglecting to include the condition f(1_R) = 1_S would cause several of the properties below to fail.

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 (also known as pseudo-rings, or non-unital rings), then the natural notion[7] is that of a rng homomorphism, defined as above except without the third condition f(1_R) = 1_S. It is possible to have a rng homomorphism between (unital) rings that is not 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 homomorphisms R → S can exist.
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 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.
The identity map is a ring homomorphism (but not the zero map).
Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
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 R → 0, where 0 denotes the zero ring (the ring whose only element is zero). This says that the zero ring is a terminal object in the category of rings.

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).
The function f : Z₆ → Z₆ defined by f([a]₆) = [4a]₆ is a rng homomorphism (and rng endomorphism), with kernel 3Z₆ and image 2Z₆ (which is isomorphic to Z₃).
There is no ring homomorphism Z_n → Z for n ≥ 1.
The complex conjugation C →C is a ring homomorphism (in fact, an example of a ring automorphism.)
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] which 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).
A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear.

Non-examples

Given a product of rings $S=R_{1}\times R_{2}$ , the natural inclusion $R_{1}\to S,x\mapsto (x,0)$ is not a ring homomorphism (unless $R_{2}$ is zero); this is because the map does not send the multiplicative identity of $R_{1}$ to that of $S$ , namely $(1,1)$ .

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 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 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 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.

Notes

Artin, p. 353
Atiyah and Macdonald, p. 2
Bourbaki, p. 102
Eisenbud, p. 12
Jacobson, p. 103
Lang, p. 88
Hazewinkel et al. (2004), p. 3. Warning: They use the word ring to mean rng.

References

Michael Artin, Algebra, Prentice-Hall, 1991.
Michael F. Atiyah and Ian G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969.
Nicolas Bourbaki, Algebra I, Chapters 1-3, 1998.
David Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1995.
Michiel Hazewinkel, Nadiya Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
Nathan Jacobson, Basic algebra I, 2nd edition, 1985.
Serge Lang, Algebra 3rd ed., Springer, 2002.

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