1
$\begingroup$

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?

$\endgroup$
1
  • 3
    $\begingroup$ In $\mathbb{Z}$, every element but $\pm 1$ is neither a unit nor a zero divisor. $\endgroup$ Dec 14, 2014 at 15:38

1 Answer 1

3
$\begingroup$

Units are always non zero-divisors. If $x$ is a unit, then $1=xy$ for some $y\in R$. Now, if $rx=0$, then $r=r(xy)=(rx)y=0$. So $x$ is not a zero-divisor.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.