# Prove that every non-zero element of a finite commutative ring with unity is either a zero-divisor or a unit. [duplicate]

Let R be a finite commutative ring with unity. Prove that every non-zero element of R is either a zero-divisor or a unit. What happens if we drop the "finite" condition on R ?

## marked as duplicate by dfeuer, tomasz, Old John, Namaste, mrfDec 6 '13 at 22:07

• For the infinite case: Look at $\mathbb Z$, what are its zero divisors? Its units? – martini Dec 6 '13 at 21:32
• I don't understand why this question have 3 votes down? +1 – Valerin Dec 6 '13 at 21:38
• Guys, look at the user's history. Pretty much we are doing the OPs homework... – LASV Dec 6 '13 at 21:39
• @LuisValerin: See Luis's comment. – tomasz Dec 6 '13 at 21:40
• Is true you are right. – Valerin Dec 6 '13 at 21:43