# A local ring with a unique minimal ideal

Let $$R$$ be a commutative ring with 1 such that it is local with a unique non-idempotent minimal ideal, that is, there are two ideals $$I$$ and $$m$$ of $$R$$ such that for each ideal $$K$$ of $$R$$ with $$0\not=K\not=R$$ we have $$I\subseteq K\subseteq m$$ and $$I^2=0$$. I am looking for a characterization for such a ring.

• Just a comment: among commutative Artinian local rings, the ones with unique minimal ideals are the quasi-Frobenius rings. – rschwieb Aug 5 at 14:07
• In case it helps to complement Sandor's answer, I know of three examples of non Noetherian rings satisfying your conditions. Two are uniserial, one is not. – rschwieb Aug 5 at 14:13

It seems to me that if $$R$$ is Noetherian, then this implies that it is an Artinian Gorenstein ring and conversely the condition holds for those. Here is why:
By your condition $$I\subseteq \mathfrak m^n$$ for every $$n$$ for which $$\mathfrak m^n\neq 0$$, but by the Krull intersection theorem the intersection of all of those is zero which implies that there must be an $$n$$ for which $$\mathfrak m^n= 0$$. This implies that $$R$$ is Artinian.
Your condition also implies that then $$I$$ is the socle of $$R$$ and it has to have dimension $$1$$ as a vector space over the residue field of $$R$$. That implies that $$R$$ is Gorenstein.
Conversely, the socle of an Artinian Gorenstein ring has dimension $$1$$ as a vector space over the residue field of $$R$$ and it is an essential submodule of $$R$$ which implies both of your conditions.