Notes on weakly-semisimple rings

Abstract

Responding to a question on right weakly semisimple rings due to Jain, Lopez-Permouth and Singh, we report the existence of a non-right-Noetherian ringRfor which every uniform cyclic right it-module is weakly-injective and every uniform finitely generated rightR-module is compressible. We show that a ringRis a right Noetherian ring for which every cyclic rightR-module is weaklyR-injective if and only ifRis a right Noetherian ring for which every uniform cyclic rightR-module is compressible if and only if every cyclic rightR-module is compressible. Finally, we characterise those modulesMfor which every finitely generated (respectively, cyclic) module in σ[M] is compressible.