Divisibility classes are seldom closed under flat covers

TL;DR

This paper investigates whether divisibility classes of modules are closed under flat covers, revealing that such closure is rare and characterizes when it occurs in terms of properties of the underlying ring.

Contribution

It establishes that divisibility classes are seldom closed under flat covers and characterizes this closure in terms of the ring being almost perfect or satisfying specific properties.

Findings

Divisibility classes are closed under flat covers iff the ring is almost perfect.

Closure under flat covers for s-divisible modules implies restrictive properties of R/sR.

Closure under flat covers is generally rare for divisibility classes.

Abstract

It is well-known that a class of all modules, which are torsion-free with respect to a set of ideals, is closed under injective envelopes. In this paper, we consider a kind of a dual to this statement - are the divisibility classes closed under flat covers? - and argue that this is seldom the case. More precisely, we show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring R/sR satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.