The Jacobson radical of monoid-graded algebras

Abstract

It is always a pleasant surprise to find that certain well-known results seemingly of a different nature can be obtained as a consequence of a general approach which absorbs and unifies all the existing methods. This right viewpoint is often the main difficultyin any subject. It certainly applies to the topic under consideration here, which is the study of the Jacobson radical of monoid-graded algebras. These algebras include such classicalobjects as group-graded algebras, crossed products, twisted monoid rings, skew monoid rings, polynomial rings, skew polynomial rings, etc. The correct approach which we shall adopt is to consider the graded radical of a module and its important special case, namely, the graded Jacobson radical of a graded algebra. A detailedaccount of all relevant background for group-graded algebras can be found in NSsta'sescu and Van Oystaeyen (1982a). As examples of successful applications of graded radicals we mention the works of NastSsescu (1984), NSst&sescu and Van Oystaeyen (1982b) and Jespers and Puczylowski (1990). The purdose of this paper is to prove a number of general results concerning the Jacobson radical of monoid-graded and group-graded algebras. One of the main theorems provides a large class of groups G for which any G-graded algebra has the property that its Jacobson radical is a graded ideal. We also demonstrate that most of what is known concerning the Jacobson radical of polynomial rings and skew polynomial rings is an easy consequence of our results.