Invertible elements in a commutative ring with no zero divisors

[Lightf]

Homework Statement

Suppose that a commutative ring R, with a unit, has no zero divisors. Does that necessarily imply that every nonzero element of R is invertible?

Homework Equations

The Attempt at a Solution

We have to show that there exists some b in R such that ab = e. Having no zero divisors implies that if b\neq0 then ab\neq0.

To show that every nonzero element of R is not invertible we must find a case where ab = c for some c in R and c \neq e.

The question seems easy but I can't wrap my head around how to write it down.

 

[voko]

Can you think of a particular example of such a ring? Is every non-zero element in it invertible?

 

[Lightf]

Oh can we just give an example, the integers form such a ring but the only invertible elements are 1 and -1. Therefore every non-zero element is not invertible and the question false.

Thanks!

 

[voko]

A counter-example is a perfectly good way to show that some particular statement is not true.