[Math] Maximal ideal in a local artinian ring.

Question

I know that an artinian ring $A$ is the union of its units and its zero-divisors.

So every non-zero-divisor is an unit.

I also know that in a local ring every element which is out from the maximal ideal is an unit.

Can I conclude that the set of zero-divisors is the maximal ideal of $A$?

Answer 1

Yes. In any commutative ring, the set of zero-divisors is the union of the prime ideals in $\operatorname{Ass}A$. An artinian ring has Krull dimension $0$, hence a local artinian ring has only $1$ prime ideal, and $\operatorname{Ass}A=\operatorname{Max}A=\operatorname{Spec}A$ is the set of zero-divisors in $A$.