Relative copure injective and copure flat modules.
Let \(R\) be a ring. The notation \(I_n\) (resp. \(F_n\)) is used for the class of all left (resp. right) \(R\)-modules of injective (resp. flat) dimension at most \(n\). A left \(R\)-module \(M\) (resp. right \(R\)-module \(F\)) is defined to be \(n\)-copure injective (resp. \(n\)-copure flat) if \(\text{Ext}^1(N,M)=0\) (resp. \(\text{Tor}_1(F,N)=0\)) for any \(N\) in \(I_n\). Clearly \(0\)-copure injectivity (resp. 0-copure flatness) is exactly copure injectivity (resp. copure flatness) which has been widely studied. The concepts are also noted to coincide with Gorenstein injectivity and Gorenstein flatness over an \(n\)-Gorenstein ring \(R\) (resp. left and right Noetherian ring with \(\text{id}(_RR)\leq n\) and \(\text{id}(R_R)\leq n)\). \(n\)-copure injectives are characterized as the kernels of \(I_n\)-precovers of injective modules. If \(S\) is a simple ring, the concepts of \(n\)-copure injectivity and flatness are respectively characterized in terms of preservation under \(\Hom(F,-)\) and \(F\otimes-\) for all flat \(R\)-modules \(F\). If \(R\) is a left coherent ring and \(M\) a finitely presented right \(R\)-module, then \(M\) is \(n\)-copure flat if and only if \(M\) is the cokernel of an \(F_n\)-pre-envelope \(K\to F\) of a right \(R\)-module \(K\) with \(F\) flat. Left IF and QF rings are characterized with respect to \(n\)-copure injectivity or \(n\)-copure flatness of certain modules. For left Noetherian rings with \(\text{id}(_RR)\leq n\), it is shown that \(\text{l\,D}(R)<\infty\) if and only if every \(n\)-copure injective module is injective or if and only if every \(n\)-copure flat right \(R\)-module is flat. The paper closes with a section on copure injective covers and copure flat envelopes. Some results are found for specific rings, for example, for a commutative Artinian ring \(R\), \(\text{id}(R)\leq 1\) is shown to be equivalent to the existence of epic copure flat envelopes or to the existence of monic copure injective covers.
scientific article; zbMATH DE number 5121400
scientific article