EMBEDDING MODULES IN GRADED MODULES OVER A SEMIGROUP-GRADED RING

We analyze conditions under which an (ungraded) module over a semigroup-graded ring can be embedded in a graded module. We use this information to provide some sufficient conditions under which the graded Jacobson radical of the ring is contained in the (ungraded) Jacobson radical. These include, and give an alternate approach to, the case of group-graded rings.