A ring homomorphism is a mapping from one ring to another that preserves the two ring operations, addition, and multiplication. This means that if R and S are rings and φ is a ring homomorphism from R to S, then for all a and b in R, the following must hold:
- φ(a + b) = φ(a) + φ(b)
- φ(ab) = φ(a)φ(b)
A ring homomorphism is called a ring isomorphism if it is both one-to-one and onto.
Properties of Ring Homomorphisms
Let φ be a ring homomorphism from R to S. The following are some important properties of ring homomorphisms:
- For any element r in R and any positive integer n, φ(nr) = nφ(r) and φ(r^n) = (φ(r))^n.
- The set φ(A) = {φ(a) | a ∈ A} is a subring of S for any subring A of R.
- If A is an ideal and φ is onto S, then φ(A) is an ideal of S.
- The set φ^-1(B) = {r ∈ R | φ(r) ∈ B} is an ideal of R for any ideal B of S.
- If R is commutative, then φ(R) is also commutative.
- If R has a unity 1, S is not equal to {0}, and φ is onto, then φ(1) is the unity of S, and units in R map to units in S.
- φ is an isomorphism if and only if φ is onto and the kernel of φ, Ker φ = {r ∈ R | φ(r) = 0}, is equal to {0}.
- If φ is an isomorphism from R onto S, then φ^-1 is an isomorphism from S onto R.
Theorem: Kernels are Ideals
Let φ be a ring homomorphism from R to S. Then, the kernel of φ, Ker φ = {r ∈ R | φ(r) = 0}, is an ideal of R.
Theorem: First Isomorphism Theorem for Rings
Let φ be a ring homomorphism from R to S. Then, the mapping from R/Ker φ to φ(R), given by r + Ker φ → φ(r), is an isomorphism. In symbols, R/Ker φ ≈ φ(R).
Theorem: Ideals are Kernels
Every ideal of a ring R is the kernel of a ring homomorphism of R. In particular, an ideal A is the kernel of the mapping r → r + A from R to R/A.
This homomorphism from R to R/A is called the natural or canonical homomorphism from R to R/A.