nonuniform local ring with essential socle

Question

I need an example: Is there any local non uniform ring $R$ with essential socle and its socle is exactly direct sum of ALL minimal left ideals? Thanks.

Answer 1

There is no such example.

That is, let $R$ be a local ring with essential left socle which is the direct sum of all minimal left ideals.

Since $R$ is local, all the minimal left ideals of $R$ are isomorphic to $R/M$ where $M$ is the maximal ideal.

Suppose $\phi :L_1\to L_2$ is an isomorphism of two such minimal left ideals. Then necessarily $L_1\oplus L_2$ is contained in the socle. But $\{x+\phi(x)\mid x\in L_1\}$ is another left ideal of $R$ which is easily seen to be isomorphic to $L_1$ and $L_2$. Including this copy of $L_1$ in the sum would make it fail to be a direct sum.

So, the left socle is necessarily just one minimal left ideal. Since you have also assumed it is essential, it is in fact contained in every other left ideal, making the ring left uniform.