Ring $R_{ 142 }$

Description:

Let $k$ be a countable field and $C$ be the ring of $k$ linear transformations of a countably infinite dimensional $k$ vector space. Let $S$ be the socle of $C$, which is the unique nontrivial ideal. Let $B$ be the subring of $C$ generated by $S$ and the identity of $C$. The required ring is $B\otimes_kB$

Keywords endomorphism ring subring tensor product