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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 $I \subset K$, 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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