The Existence of Flat Covers

Abstract

We show that over a right coherent ring all pure injective left modules have flat covers. Then using recent work of Auslander and Buchweitz we show that left modules of finite flat dimension over right coherent rings also have flat covers.

The existence of Gorenstein flat covers

“…If R has finite global dimension, then F is the class of flat modules and so F -covers are in fact flat covers whose existence was shown in [2]. Finally, if R is a Gorenstein ring, then F becomes the class of Gorenstein flat modules (cf.…”

Coherent rings of finite weak global dimension

“…envelope) is still an open problem. Nevertheless, we know that flat covers exist over right coherent rings of finite weak global dimension [3] and, restricting the value of that dimension to at most two, we have also guaranteed the existence of flat envelopes with the additional property that the associated diagrams can be completed in a unique way [2].…”

“…When F is the class of all flat left R-modules, the objects of F ⊥ are the cotorsion modules as defined in [7]; these were later used by Belshoff, Enochs and Xu in the determination of two wide classes of rings over which each module has a flat cover ( [3], [14]). The relevant class of modules in this paper, however, is a subclass of ⊥ F which is intimately related with flat envelopes.…”

“…In Section 2, we use some properties of these covers and envelopes to study the modules M such that Ext [3], [14]), and our main concern in this paper will be with the condition Ext 1 R (M, F) = 0 and its relation to flat envelopes. In this way, we consider in Section 3 the modules M such that Hom R (M, F) = 0 and Ext 1 R (M, F) = 0 (called ϕ-torsion modules).…”

Structure of flat covers of injective modules

“…This question has been studied by several authors; see for example [1,2,12]. Recently, Bican, El Bashir and Enochs have proved that all modules have flat covers (see [3]).…”