Semiprimitivity of Group Algebras: a Survey
نویسندگان
چکیده
Let K be a field and let G be a multiplicative group. The group ring K[G] is an easily defined, rather attractive algebraic object. As the name implies, its study is a meeting place for two essentially different algebraic disciplines. Indeed, group ring results frequently require a blend of group theoretic and ring theoretic techniques. A natural, but surprisingly elusive, group ring problem concerns the semiprimitivity of K[G]. Specificially, we wish to find necessary and sufficient conditions on the group G for its group algebra to have Jacobson radical equal to zero. More generally, we wish to determine the structure of the ideal JK[G]. In the case of infinite groups, this problem has been studied with reasonable success during the past 40 years, and our goal here is to survey what is known. In particular, we describe some of the techniques used, discuss a number of the results which have been obtained, and mention several tantalizing conjectures.
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