Projective indecomposable modules and quivers for monoid algebras

We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, &-trivial and Script capital R-trivial monoids and left regular bands of groups.