Abstract
A left R-module M is called almost F-injective, if every R-homomorphism from a finitely presented left ideal to M extends to a homomorphism of R to M. A right R-module V is said to be almost flat, if for every finitely presented left ideal I, the canonical map V ⊗ I → V ⊗R is monic. A ring R is called left almost semihereditary, if every finitely presented left ideal of R is projective. A ring R is said to be left almost regular, if every finitely presented left ideal of R is a direct summand of RR. We observe some characterizations and properties of almost F-injective modules and almost flat modules. Using the concepts of almost F-injectivity and almost flatness of modules, we present some characterizations of left coherent rings, left almost semihereditary rings, and left almost regular rings.
Highlights
Throughout this paper, R denotes an associative ring with identity and all modules considered are unitary
A left R-module M is called n-presented [2] if there is an exact sequence of left R-modules Fn → Fn−1 → · · · → F1 → F0 → M → 0 in which every Fi is a finitely generated free, equivalently, projective left R-module
A left R-module M is called F-injective [4] if every R-homomorphism from a finitely generated left ideal to M extends to a homomorphism of R to M, or equivalently, if Ext1(R/I, M) = 0 for every finitely generated left ideal I
Summary
Throughout this paper, R denotes an associative ring with identity and all modules considered are unitary. A left R-module M is said to be FP-injective [9] if Ext1(A, M) = 0 for every finitely presented left R-module A. A left R-module M is called F-injective [4] if every R-homomorphism from a finitely generated left ideal to M extends to a homomorphism of R to M, or equivalently, if Ext1(R/I, M) = 0 for every finitely generated left ideal I. We recall that a ring R is called left coherent if every finitely generated left ideal of R is finitely presented. We shall call a ring R left almost semihereditary if every finitely presented left ideal of R is projective. Left almost semihereditary rings and left almost regular rings will be characterized by almost F-injective left R-modules and almost flat right R-modules
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