Ring homomorphism

In mathematics, 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 $f:R\to S$ that preserves addition, multiplication and multiplicative identity; that is,

${\begin{aligned}f(a+b)&=f(a)+f(b),\\f(ab)&=f(a)f(b),\\f(1_{R})&=1_{S},\end{aligned}}$

for all $a,b$ in $R.$

These conditions imply that additive inverses and the additive identity are preserved too.

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 have exactly the same properties.

If R and S are rngs, then the corresponding notion is that of a rng homomorphism, 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 rings forms a category with ring homomorphisms as morphisms (see 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 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 a two-sided ideal in R. Every two-sided ideal in a ring R is the kernel of some ring homomorphism.
• A homomorphism is injective if and only if kernel is the zero ideal.
• 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.
• 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.

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

Main article: 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.

See also

• Change of rings

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