Is there a non-artinian noetherian ring whose non-units are zero-divisors? 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.]
 A: 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
A: This is a community wiki answer whose purpose is to rephrase R. C. Cowsik's beautiful answer with a slightly different notation.
Here is an example of a non-artinian noetherian ring $A$ whose non-units are zero-divisors.
Let $K$ be a field, $X$ an indeterminate, and $A$ the additive group $K[[X]]\times K$. One  checks that the formula 
$$
(f,\lambda)(g,\mu)=(fg,\lambda g(0)+\mu f(0))
$$ 
defines on $A$ a structure of commutative ring with one. [See this answer of user26857 for a generalization of this construction.] 
One verifies that $A$ is noetherian, that $A$ has exactly two prime ideals, namely $\mathfrak p:=(0)\times K$ and $\mathfrak m:=(X)\times K$, that they satisfy $\mathfrak p\subset\mathfrak m$, and that we have $(f,\lambda)(0,1)=(0,0)$ for all $(f,\lambda)\in\mathfrak m$.
This shows that $A$ is indeed a non-artinian noetherian ring whose non-units are zero-divisors.
