# Prove that every nonzero element of R is either a unit or a zero–divisor. [duplicate]

Let $R$ be a finite commutative ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero–divisor.

Sol:

Let $a\not=0$

Because $R$is finite then

$a^j=a$ , then $(a^j -a )=0$

$a (a^{j-1}-1) =0$

If $a\not=0$ then $a$ is zero divisor and $a^j a^{-1} = a^{j-2}a=1$ so $a$ is unit

is true to prove this theorm by this way ? ,if not what is true ? Thanks for all

• But , my prove is it true ?because l am little bit confused ? – user508518 Mar 27 '18 at 11:01
• I don't think this is a duplicate. The OP here asks specifically if his/her own proof works or not. – user1551 Mar 27 '18 at 11:28
• @user1551 and a logical consequence of that line of thinking is that you create a loophole through which arbitrarily many duplicates can be made. If this person had a solution to a problem posted on this site, then the advisable thing to do is to post it as a solution. – rschwieb Mar 27 '18 at 11:38
• math.stackexchange.com/a/2710194/29335 – rschwieb Mar 27 '18 at 16:10