Let be a ring with . A nonzero element is called a left zero divisor in if there is a nonzero element such that . Symmetrically, is called a right zero divisor in if there is a nonzero element such that . (So a zero divisor is an element which is either a left or a right zero divisor, or both.) An element is said to have a left inverse in if there is some such that . Symmetrically, has a right inverse if there exists such that .
- Prove that is a unit if and only if it has both a right and a left inverse.
- Prove that if has a right inverse then is not a right zero divisor.
- Prove that if has more than one right inverse then is a left zero divisor.
- Prove that if is a finite ring then every element that has a right inverse is a unit (i.e. has a two sided inverse).
- First, suppose is a unit. Then there exists such that ; thus is both a left and a right inverse for . Suppose now that and . (That is, that is a right inverse of and is a left inverse.) Now , so that . Thus has a two sided inverse, and thus is a unit.
- Suppose has a right inverse and is a right zero divisor; that is, for some and for some , where . Then implies , so that , a contradiction. So if an element has a right inverse, it is not a right zero divisor.
- Suppose has distinct right inverses and – that is, , and . Now and , but , so that is a left zero divisor.
- Let be a finite ring. Let have a right inverse; say . Now define by . Suppose ; then , so that , and we have . Thus is injective. Since is finite, in fact is surjective. Thus for some , and we have . Since has a left and right inverse, by part 1 above, is a unit in .