Modules Sharing Common Flat Precovers
by
Karen D. Akinci
Dokuz Eylul Uni. Izmir Turkey
Coauthors: Refail Alizade (Dokuz Eylul Univ)

Given an associative ring R and a class of (left) R-modules X we can define, by Enochs [3], [5], an X-cover and an X-precover of an R-module, and its X-envelope and X-preenvelope. The existence question where X is replaced by a specific class has been extensively studied by Auslander and Rieten, Enochs [3] and others. It has long been known for example that where X is the class of injective modules an X-envelope exists for every R-module, simply the well known injective hull of that module. Bass solved the existence problem for projective covers in the 1960's concluding that there is a class of rings, the perfect rings, in which a projective cover exists for every R-module. If we specify the class X to be that of the flat left R-modules the existence question on X-covers becomes the flat cover conjecture of Enochs and has recently been proved in the affirmative [2], [4].

It is well known [1], that if g:M --> N is a superfluous epimorphism, then f:P --> M is a projective cover for M, if and only if f o g:P --> N is a projective cover for N. We focus our attention on the simillar question of the effect of a flat cover of a certain module on another module connected to it via an epimorphism, the situation where two modules share a common flat cover.

Theorem 1 If there exists a flat cover f:F --> M and an epimorphism p:M --> N then Kerp being cotorsion implies that F is also a flat precover of N.

The converse of this theorem does not necessarily hold, however with a logical constriction there is a valid converse in the form of the following theorem.

Theorem 2 Let f:F --> M and p:M --> N be as in Theorem 1, if p o f:F --> N is a flat (pre)cover then Kerp is cotorsion.

On the other hand there is a very interesting question converse to the above. That is if F is a flat cover of N and there exists an epimorphism form M to N then is there a connection between the flat cover of N and the R-module M? This is answered in the affirmative below for the case when the ring R is such that the cotorsion modules are closed under submodules, that is, satisfying the cotorsion condition CC.

Theorem 3 Let R be a ring satisfying CC If f:F --> N is a flat cover and p:M --> N is a superfluous epimorphism then Kerp is cotorsion implies that F is a flat pre-cover of M.

References

[1] Anderson F.W. & Fuller K.R. (1974), Rings and categories of modules, volume 13 of Graduate Texts in Mathematics. Springer-Verlag.

[2] Bican L. & El Bashir R. (1999). Over every ring, every module has a flat cover. (preprint)

[3] Enochs E. (1981). Injective and flat covers, envelopes and resolvents. Israel J.Math., 39:189-209.

[4] Trlifaj J. (1999). A generalisation of the flat cover conjecture. (preprint)

[5] Xu J. (1996). Flat covers of modules. Springer-Verlag.

Date received: May 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-07.