Covering classes and uniserial modules

2020/01/09 by Facchini, Alberto, Nazemian, Zahra, Prihoda, Pavel
#16S50. Secondary 16W80 #FOS: Mathematics #Primary 16P70 #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2001.03085

Abstract

We apply minimal weakly generating sets to study the existence of Add(U_R)-covers for a uniserial module U_R. If U_R is a uniserial right module over a ring R, then S:=End (U_R) has at most two maximal (right, left, two-sided) ideals: one is the set I of all endomorphisms that are not injective, and the other is the set K of all endomorphisms of U_R that are not surjective. We prove that if U_R is either finitely generated, or artinian, or I ⊂ K, then the class Add(U_R) is covering if and only if it is closed under direct limit. Moreover, we study endomorphism rings of artinian uniserial modules giving several examples.

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• Enochs' conjecture for small precovering classes of modules
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