Abstract Algebra/Ring Homomorphisms
Just as with groups, we can study homomorphisms to understand the similarities between different rings.
Let R and S be two rings. Then a function is called a ring homomorphism or simply homomorphism if for every , the following properties hold:
In other words, f is a ring homomorphism if it preserves additive and multiplicative structure.
Furthermore, if R and S are rings with unity and , then f is called a unital ring homomorphism.
- Let be the function mapping . Then one can easily check that is a homomorphism, but not a unital ring homomorphism.
- If we define , then we can see that is a unital homomorphism.
- The zero homomorphism is the homomorphism which maps ever element to the zero element of its codomain.
Theorem: Let and be integral domains, and let be a nonzero homomorphism. Then is unital.
Proof: . But then by cancellation, .
In fact, we could have weakened our requirement for R a small amount (How?).
Theorem: Let be rings and a homomorphism. Let be a subring of and a subring of . Then is a subring of and is a subring of . That is, the kernel and image of a homomorphism are subrings.
Proof: Proof omitted.
Theorem: Let be rings and be a homomorphism. Then is injective if and only if .
Proof: Consider as a group homomorphism of the additive group of .
Theorem: Let be ﬁelds, and be a nonzero homomorphism. Then is injective, and .
Proof: We know since fields are integral domains. Let be nonzero. Then . So . So (recall you were asked to prove units are nonzero as an exercise). So .
Let be rings. An isomorphism between and is an invertible homomorphism. If an isomorphism exists, and are said to be isomorphic, denoted . Just as with groups, an isomorphism tells us that two objects are algebraically the same.
- The function defined above is an isomorphism between and the set of integer scalar matrices of size 2, .
- Similarly, the function mapping where is an isomorphism. This is called the matrix representation of a complex number.
- The Fourier transform defined by is an isomorphism mapping integrable functions with pointwise multiplication to integrable functions with convolution multiplication.
Exercise: An isomorphism from a ring to itself is called an automorphism. Prove that the following functions are automorphisms:
- Define the set , and let