  Let $R$ be a commutative Noetherian local ring and $M$ a finitely generated
$R$-module. We introduce a general form of the classically studied trace map
that unifies several notions from the literature. We develop a theory around
these objects and use it to provide a broad extension of a result of Lindo
calculating the center of $\operatorname{End}_R(M)$. As a consequence, we show
under mild hypotheses that in dimension $1$, the canonical module of
$Z(\operatorname{End}_R(M))$ may be calculated as the trace submodule of $M$
with respect to the canonical module of $R$.
