Searching for a certain local ring

Question

I am searching for a local ring $R$ such that $R/\mathrm{Soc}(R_R)$ is not isomorphic to the two-element ring $\mathbb Z_2$.

Here, by $\mathrm{Soc}(R_R)$ I mean the socle of the right $R$-module $R$.

Thanks for any help!

Answer 1

If you take $K$ to be any field other than $\mathbb{Z}_2$, then the ring $R=K[X]/(X^2)$ satisfies your requirements, since it is local and $R/\mathrm{Soc}(R_R) \cong K$.

Answer 2

For $p$ a prime and $n>2$, the only minimal ideal of $\mathbb{Z}/p^n\mathbb{Z}$ is $p^{n-1}\mathbb{Z}/p^n\mathbb{Z}$ and the quotient ring modulo the socle is $\mathbb{Z}/p^{n-1}\mathbb{Z}$: it is a uniserial ring, that is, its ideals are linearly ordered.

For $p\ne2$ the example $\mathbb{Z}/p^2\mathbb{Z}$ is even simpler, because the socle is the same as the maximal ideal.