Structure theory of simple rings without finiteness assumptions

Abstract

It is the purpose of this paper to lay the foundations of a general theory of simple rings, both associative and non-associative. In part I we obtain the structure of simple associative rings that either contain minimal right ideals or contain maximal right ideals. In either case we obtain a realization of our ring 2f as a certain type of ring of linear transformations in a vector space over a division ring. If 2{ has a minimal right ideal this realization is essentially unique and we can prove a converse theorem that rings of linear transformations having certain properties are simple and contain minimal right ideals. Thus the theory here is essentially as complete as that of the classical case of rings that satisfy the descending chain condition for right ideals. Our structure theory hinges on a general theorem (Theorem 6) on the structure of irreducible rings of endomorphisms of commutative groups. This theorem may be regarded as an extension of Burnside's theorem on irreducible algebras of matrices. The classical theory of simple rings with descending chain condition is a simple consequence of our results and we believe that the present treatment is more transparent than the methods of proof previously given(1) . In studying an arbitrary non-associative ring 9t one is led to consider the associative ring 92 generated by the left and the right multiplications x->ax and x-xcxa acting in 2. If 21 is simple, 91 is an irreducible ring of endomorphisms. In part II we describe the structure of $1 in terms of the multiplication centralizer ( of 2[ defined to be the totality of endomorphisms 'y in 2 such that (xy)y=(xy)y=x(yy). We define the concepts of center, central algebra and extension of the underlying field of an algebra. In Part III we investigate a special type of simple associative algebra that may be regarded as a generalization of the concept of the complete algebra of linear transformations in a vector space over a field, or equivalently, of the concept of the complete matrix algebra. There are many points of contact between the discussion here (and in other parts of the paper) and recent work in the theory of rings of transformations in Banach spaces and in vector spaces over the fields of reail and complex numbers(2).