Compressible Modules And Weakly Compressible Modules

We introduce the notions of right compressible module and weakly compressible module property in a general module theoretic setting.Let R be a ring and M be a right R-module.If for every non-zero submodule N of M there exists a monomorphism from M to N,then M is called compressible; if for every non-zero submodule N of M there exists a non-zero homomorphism from M to N such that f~2≠0,then M is called weakly compressible. In this paper,we give some characterizations of compressible module and weakly compressible module,respectively,and show that a direct summand of compressible module the property,and a direct sum of weakly compressible module inherit the property.Let R be a right Noetherian ring and M be a right R-module,then M is compressible if and only if there exists a non-zero monomorphism from M to U,for every uniform submodule U of M.Over a right Noetherian ring R,we show that,for a non-zero bounded right R-module M,there exists a monomorphism f:M→R/P for every associated prime ideal P of M if and only if exists a monomorphism g:M→M' for every ralated right R-module M'.Finally,Let R be any ring and let M be a non-singular injective right R-module such that m R has finite rank for every element m of M,then M is weakly compressible if and only if M is semisimple. Among other results,we also show that every non-zero right R-module is weakly compressible over a right semi-artinian right V-ring.