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.


  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:

  2. Define the set  , and let