Copure Projective Modules And The Monomorphic Flat Envelopes Of Modules

In this thesis copure projective modules are characterized and the monomor-phic flat preenvelopes and the flat envelopes of modules are studied. Let R be a ring. An R-module M is called a copure projective module if Ext1/R(M, F)=0for any flat module F. It is proved that if M is copure projective, then either M is projective or fdR-M≥2. The concept of CPH rings which means that sub-modules of a copure projective module are copure projective is introduced in this thesis. It is proved that a ring R. is a CPH-ring if and only if the submodules of projective modules are copure projective; if and only if the injective dimension of flat modules is at most one:if and only if every ideal of R are copure projective.Besides, the monomorphic flat preenvelopes and monomorphic flat envelopes of modules are also studied in this thesis. It is proved that if F is the monomorphic flat envelope of M, then F is a quotient copure projective essential extension of M. It is also proved that if R is a domain and a flat module F is a torsion quotient copure projective extension of M, then F is a monomorphic flat envelope of M.