A short note on strongly flat covers of acts over monoids

Abstract

Recently two different concepts of covers of acts over monoids have been studied. That based on coessential epimorphisms and that based on Enochs' definition of a flat cover of a module over a ring. Two recent papers have suggested that in the former case, strongly flat covers are not unique. We show that these examples are in fact false and so the question of uniqueness appears to still remain open. In the latter case, we re-present an example due to Kruml that demonstrates that, unlike the case for flat covers of modules, strongly flat covers of S−acts do not always exist.

Covers of Finitely Generated Acts over Monoids

“…In Section 5 we study Enochs' notion of cover in the category of acts over monoids and focus on X -precovers, where X is a class of S-acts closed under isomorphisms. Basic results on covers of acts over monoids can be found in [3,4,6,11].…”

Covers of Finitely Generated Acts over Monoids

“…Subsequently, several additional papers concerning covers have appeared (e.g., [7][8][9][10]). Recently, Irannezhad and Madanshekaf [11] researched covers of S-acts over monoids with Condition (P ′ ), and Qiao et al introduced Condition (PF ′′ ) covers in a similar way in [12].…”