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ABSTRACT
Let be a commutative ring and
an
-module. Then
is a multiplication module if
for each submodule
of
. The ideal
of
has proved useful in studying multiplication modules. We show that if
is a faithful multiplication module, then
an ideal of
, the trace ideal of
. Moreover,
is an idempotent multiplication ideal of
and
. We also show that for a multiplication module
,
is an ideal of the endomorphism ring
of
and that
where the inverse limit is taken over the finitely generated submodules
of
.