Throughout is a ring with 1 and is a unitary left module. An module homomorphism of is called an endomorphism of . The set of endomorphisms of is denoted by or See that is a ring, where is the function composition.

**Example 1**.

*Proof*. Define by for all It is easy to see that is a ring homomorphism. It is one-to-one because if and only if for all So if we let we’ll get It is onto because if then letting we’ll have and thus

**Theorem 1**. let and suppose is the set of all matrices with Then

*Proof*. For every define and by and Now define by Then

1) is well-defined : for all and thus

2) is a homomorphism* *: let Then clearly holds. Also, since we have

3) is injective* *: because for all

4) is onto* *: for any let See that

**Remark**. If in the above theorem then We will also have for all and thus Therefore we get this important result: