Linearly compact rings and selfcogenerators (Q760478)

A module M is called strongly quasi-injective, if for every submodule N, every homomorphism f:N\(\to M\) and any \(x\not\in N\) there exists a homomorphism \(\hat f\) of M, which extends f with \(\hat f(\)x)\(\neq 0\); a module M is called \(\Sigma\)-strongly quasi-injective, if the direct sum of any family of copies of M is a strongly quasi-injective module. It is proved, that the ring A admits a right linearly compact topology iff there exists a ring R and a strongly quasi-injective module \({}_ RK\) with essential socle, every finitely generated submodule of which is linearly compact in the discrete topology, such that \(A=End(_ RK)\). Moreover, A admits a right strongly linearly compact topology iff \({}_ RK\) is a \(\Sigma\)-strongly quasi-injective module with an essential socle.