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][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 $f\colon R\rightarrow S$ 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 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 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.

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 $\rho :k\to \operatorname {End} (V)$ given by $\rho (a)v=av$ 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 $R\to \operatorname {End} (M)$ .
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 $R\to R\times S$ 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 $R\times S$ .

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.

Citations

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.

Notes

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.

References

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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[FOOTNOTEArtin1991353-1] Artin 1991, p. 353.

[FOOTNOTEAtiyahMacdonald19692-2] Atiyah & Macdonald 1969, p. 2.

[FOOTNOTEBourbaki1998102-3] Bourbaki 1998, p. 102.

[FOOTNOTEEisenbud199512-4] Eisenbud 1995, p. 12.

[FOOTNOTEJacobson1985103-5] Jacobson 1985, p. 103.

[FOOTNOTELang200288-6] Lang 2002, p. 88.

[FOOTNOTEHazewinkel20043-7] Hazewinkel 2004, p. 3.

[8] Hazewinkel initially defines "ring" without the requirement of a 1, but very soon states that from now on, all rings will have a 1.

[9] 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.