Is there a non-artinian noetherian ring whose non-units are zero-divisors?
Equivalent formulation:
Is there a noetherian ring of positive dimension whose non-units are zero-divisors?
[In this post, "ring" means "commutative ring with one", and "dimension" means "Krull dimension".]
Here is the motivation:
Let $A$ be a ring whose non-units are zero-divisors.
If $A$ is not noetherian, then $A$ can have positive dimension: see this answer of user18119.
If $A$ is noetherian and reduced, then $\dim A\le0$: see this answer of user26857.
[Recall that a noetherian ring is artinian if and only if its dimension is $\le0$. Recall also that a ring has the property that its non-units are zero-divisors if and only if it is isomorphic to its total ring of fractions.]
Best Answer
Yes. Example: Let $B = K[[X]]$, with $K$ a field. Let $M = B/(X)$, the residue module. Let $A = B \oplus M$, with product natural on $B$, action of $B$ on $M$ and $a^2 = 0$, all $a\in M$. Then $A$ is clearly noetherian, $\dim A = 1$ and hence not Artinian. The non units of $A$ are $N = (X) \oplus M$. Any element in $N$ times any element in $M$ is $0$.