Elements in a commutative ring ( Zero divisors , Unit)

Question

This question was part of my algebra quiz and I was unable to solve 2 options out of 4. So, I am asking it here for help.

Let R be a finite non-zero commutative ring with unity. Then which of the sattements are true?

Any non zero element of ring is either a unit or a zero divisor.
There may exist an element which is neither a ring nor a zero divisor.

I studied algebra from Joseph Gallian.

Can you please tell how should I approach this particular problem as I am really confused if there exists an element which is neither unit nor zero divisor.

I can't think of any idea.

Thanks

“Neither a ring nor a zero divisor” presumably meant “neither a unit”? — rschwieb, Nov 23 '20 at 13:18
To find the duplicate, I searched “finite ring unit zero divisor” and it was the top hit. — rschwieb, Nov 23 '20 at 13:20

score 1 · Accepted Answer · answered Nov 23 '20 at 13:24

It's critical here that the ring is finite. Indeed, let $a \in R$ be nonzero. If $a$ is not a zero divisor then the map $R \longrightarrow R$ via $x \mapsto ax$ is injective. As $R$ is finite, it is therefore a bijection by the Pigeonhole Principle. Hence, there is some $x \in R$ such that $ax = 1$, i.e. $a$ is a unit.

Note that this is false in general for infinite rings. For instance, $2 \in \mathbb Z$ is not a zero divisor but $1/2$ is not an integer, so 2 is not a unit in.$\mathbb Z$.

Elements in a commutative ring ( Zero divisors , Unit)

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