Properties of Modules and Rings Relative to Some Matrices

Let R be a ring and  _β×α(R) (?   _β×α(R)) the set of all β × α full (row finite) matrices over R where α and β ≥ 1 are two cardinal numbers. A left R-module M is said to be “injective relative” to a matrix A ? ?   _β×α(R) if every R-homomorphism from R ^(β) A to M extends to one from R ^(α) to M. It is proved that M is injective relative to A if and only if it is A-pure in every module which contains M as a submodule. A right R-module N is called flat relative to a matrix A ?  _β×α(R) if the canonical map μ: N? R ^(β) A → N ^α is a monomorphism. This extends the notion of (m, n)-flat modules so that n-projectivity, finitely projectivity, and τ-flatness can be redefined in terms of flatness relative to certain matrices. R is called left coherent relative to a matrix A ?  _β×α(R) if R ^(β) A is a left R-ML module. Some results on τ-coherent rings and (m, n)-coherent rings are extended.