  This paper develops the fundamentals of modular representation theory for
finite monoids, introducing the decomposition matrix and exploring its
connection to Brauer characters. We define modular characteristic and explain
how the representation theory in nonmodular positive characteristic behaves
like the characteristic zero theory by showing that one can lift from
nonmodular characteristic $p$ all simple and projective indecomposable modules,
as well as a quiver presentation of the basic algebra. As an application of the
theory developed, we give a new proof of Glover's theorem that the monoid of
$2\times 2$-matrices over $\mathbb F_p$ has infinite representation type over
fields of characteristic $p$.
  We also investigate the relationship between nonsingularity of the Cartan
matrix of a monoid algebra in characterstic zero and in positive
characteristic. We show that for von Neumann regular monoids the Cartan matrix
is always nonsingular and we show that if a monoid has aperiodic left (or
right) stabilizers, then nonsingularity in characteristic zero implies
nonsingularity in positive characteristic. Florian Eisele has recently shown
that the Cartan matrix of a monoid algebra can be nonsingular in characteristic
zero and singular in positive characteristic, disproving a conjecture of the
author in an earlier version of this paper. A new conjecture is proposed,
unifying the cases of regular monoids and monoids with aperiodic stabilizers.
