## Ring homomorphisms map an identity element to an identity or a zero divisor

Let and be nonzero rings with identies and , respectively. Let be a nonzero ring homomorphism.

- Prove that if , then is a zero divisor in . Deduce that if is an integral domain then every ring homomorphism sends the identity of to the identity of .
- Prove that if then if is a unit, then is a unit and .

- Suppose , with . First, if , then , so that , a contradiction. Thus . Now , so that , and we have . Thus is a left zero divisor. Similarly, , and is a right zero divisor. Hence is a zero divisor in .
If is an integral domain, this yields a contradiction. Thus if is a nonzero ring homomorphism and and have identities, then . (I.e. is a unital ring homomorphism.)

- Suppose is a unit. Now . Similarly, . Thus is a unit, and by the uniqueness of inverses, .

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