  The trace on matrix rings, along with the augmentation map and Kaplansky
trace on group rings, are some of the many examples of linear functions on
algebras that vanish on all commutators. We generalize and unify these examples
by studying traces on (contracted) semigroup rings over commutative rings. We
show that every such ring admits a minimal trace (i.e., one that vanishes only
on sums of commutators), classify all minimal traces on these rings, and give
applications to various classes of semigroup rings and quotients thereof. We
then study traces on Leavitt path algebras (which are quotients of contracted
semigroup rings), where we describe all linear traces in terms of central maps
on graph inverse semigroups and, under mild assumptions, those Leavitt path
algebras that admit faithful traces.
