Abstract Algebra/Ring Homomorphisms

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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.


  1. Let be the function mapping . Then one can easily check that is a homomorphism, but not a unital ring homomorphism.
  2. If we define , then we can see that is a unital homomorphism.
  3. 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.


  1. The function defined above is an isomorphism between and the set of integer scalar matrices of size 2, .
  2. Similarly, the function mapping where is an isomorphism. This is called the matrix representation of a complex number.
  3. 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:

  1. Define the set , and let