Absolutely pure modules

Abstract

A module A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A has a solution in A. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, R is Noetherian if and only if every absolutely pure i?-module is injective and semihereditary if and only if the class of absolutely pure i?-modules is closed under homomorphic images. If R is a Prüfer domain, then the absolutely pure i?-modules are the divisible modules and Ext\t(M, A)=0 whenever A is divisible and M is a countably generated torsionfree .R-module.

On a Problem of Fuchs and Salce

“…R I ≤ 1. On the other hand, left semihereditary rings are exactly those over which quotients of absolutely pure left modules are absolutely pure (see Megibben, 1970). This proves (a) ⇒ (b).…”

Remarks on f p-Injective and f p-Flat Modules

“…It is well known that R is a left semihereditary ring if and only if R is left coherent and every submodule of any flat right R-module is flat if and only if every quotient of any F P-injective left R-module is F P-injective (see [14,Theorem 2]). Combining this with [9, Theorem 2.4], we get the following observation.…”

“…It is well known that quasi-Frobenius rings can be characterized as the rings R that determine an R-duality between the categories of finitely generated left and right R-modules R. In turn, an R-duality for the categories of finitely presented left and right modules leads to the class of weakly quasi-Frobenius rings, i.e., R is a two-sided F P-injective two-sided coherent ring [9], where R is called a left coherent ring [2] if every finitely generated left ideal is finitely presented; or equivalently, every R-dual module of any finitely presented right R-module is finitely presented (see [3,Proposition 1]). A left R-module M is said to be F P-injective (or absolutely pure) [14,20] if Ext 1 (F, M) = 0 for any finitely presented left R-module F; or equivalently, M is a pure submodule in every left R-module containing M. R is called a left F P-injective ring if R is F P-injective as a left R-module.…”

n-Coherent Rings

“…By hypothesis, R I is an n-flat right R-module, thus Tor R 1 (R I , F=H) ¼ 0. The proof given by Lenzing [7] carries over without change to conclude that H is finitely presented. Hence (a) follows, completing the proof.…”