Topological Morita Equivalences Induced by Ideals Generated by Dense Idempotents,

Given two complete right linearly topologized rings (R, ρ) and (S, σ), and a bimoduleRBSendowed with a complete topology β, in such a way that (BS, β) CLT-(S, σ) and there be a continuous ring homomorphism (R, ρ) → CEnduS(B, β), we define a functor − RB:CLT-(R, ρ) → CLT-(S, σ) which is left adjoint to the functor CHomuS((B, β), −):CLT-(S, σ) → CLT-(R, ρ). Then we consider the particular case in which (S, σ) = eRewith its induced topology, whereeis a dense idempotent ofR(that is,ReRis dense in (R, ρ)). Under these hypotheses we show that the pair of functors − R Re:CLT-(R, ρ) → CLT-(S, σ) and − SeR:CLT-(S, σ) → CLT-(R, ρ) is an equivalence of categories. As an application of this result, we re-obtain a theorem of Xu, Shum, and Turner-Smith on similarities between infinite matrix subrings.