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Let $R$ be a ring. An $R$-module $M$ is called a weak generator
for a class $C$ of $R$-modules if Hom$_{R}(M, V)$ is non-zero for
every non-zero module $V$ in $C$. A projective module $M$ is a
weak generator for $C$ if and only if $M \neq MA$ for every
annihilator $A$ of a non-zero module $V$ in $C$. Given any class
$C$ of $R$-modules, a finitely annihilated $R$-module $M$ is a
weak generator for the class of injective hulls of modules in $C$
if and only if the $R$-module $R/A$ is a weak generator for $C$,
where $A$ is the annihilator of $M$. Moreover a finitely
annihilated $R$-module $M$ is a weak generator for the class of
all injective $R$-modules if and only if the annihilator of $M$ is
a left T-nilpotent ideal. In case the ring $R$ is commutative, a
finitely generated $R$-module $M$ is a weak generator for the
class of all $R$-modules if and only if $M$ is a weak generator
for the class of injective $R$-modules. In addition, if the ring
$R$ is Morita equivalent to a commutative semiprime Noetherian
ring, then $M$ is a weak generator for the class of all
$R$-modules if and only if the trace of $M$ in $R$ is an essential
right ideal of $R$.
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