Some Remarks on Algebras Over an Algebraically Closed Field

The theory of rings with radicals is an interesting and far reaching problem of modern algebra.' In this paper we have examined some aspects of algebras which may have radicals and whose coefficient fields are algebraically closed. Some of the methods employed clearly could be used for less restricted algebras, but a full extension of the results requires the solution of a number of problems still under investigation.2 The authors feel that the theory of algebras over an algebraically closed field has some interest and value in itself, particularly in view of its immediate application to the representation theory of finite groups. Moreover, this restricted case is a valuable testing ground for theorems for more general rings. In the first part of the paper we have studied the concept of basic algebra. The basic algebras are semi-primitive subalgebras (i.e. modulo their radicals are direct sums of division algebras, in our case, direct sums of fields) which for the algebras under discussion play a role in some respects analogous to that of division algebras for simple algebras. Related to the basic algebras are the Cartan basis systems,3 and systems of elementary modules.4 The commutator algebras of matrix representations of an algebra, or what is equivalent, the algebras of homomorphisms of the related representation spaces, can be analyzed in a rather simple manner. We shall say that a linear function sp of an algebra a is symmetric, if for every a, E c a, (p(af) = sp(Oa). In the case where a is over an algebraically closed field, and is also semisimple, the characters of the irreducible representations of a form a complete set of symmetric functions of a. When a has a radical, this is no longer true. In Part 2 of the paper, we discuss symmetric functions of algebras with radical. In Part 3 of the paper the regular representations are written in terms of elementary modules.