Let $R$ be a finite commutative ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
I know this question has a solution here Every nonzero element in a finite ring is either a unit or a zero divisor.
But I want you to check my proof.
Let $a \not = 0, a \in R$.
We have to prove that $a$ is either a unit or a zero divisor.
Let $a$ is a unit then we have to show that it is not a zero divisor.
1) $\exists x \in R$ such that $ax = 1.$
2) Let for some $b \in R$, $ab = 0$ is true.
Now $ax = 1$ $\implies ax + 0 = 1 \implies ax + ab = 1 \implies a(x+b) = 1 .$
$\therefore$ $x$ and $(x+b)$ both are multiplicative inverses of $a \implies x = x+b$
$\therefore$ $b = 0$ hence $a$ is not a zero divisor
In a similar way we can show that if $a$ is a zero divisor then it is not a unit.
a) Is my proof okay?
b) I am not using the "finite" and "comutative" conditions of $R$.