Let be a ring with 1, and fix an element . Let be a left -module. Prove that the mapping is an -module endomorphism of . Prove further that if is commutative, the mapping given by is a ring homomorphism.
Let , and let and . Then . Thus is an -module endomorphism of .
Now suppose is commutative and define by . Let and let . Note that . Thus . Similarly, , so that . Thus is a ring homomorphism. Note also that , so that . Thus is in fact a unital ring homomorphism.