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