Properties of Modules and Rings Relative to Some Matrices

Abstract

Let R be a ring and 𝕄 β×α(R) (ℝ 𝔽 𝕄 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 ∈ ℝ 𝔽 𝕄 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 ∈ 𝕄 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 ∈ 𝕄 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.