# Abstract Algebra/Ring Homomorphisms

Just as with groups, we can study *homomorphisms* to understand the similarities between different rings.

## Homomorphisms[edit | edit source]

### Definition[edit | edit source]

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**.

### Examples[edit | edit source]

- 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 .

## Isomorphisms[edit | edit source]

### Definition[edit | edit source]

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*.

### Examples[edit | edit source]

- 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