Algebras over infinite fields
A. S. Amitsur · Proceedings of the American Mathematical Society · 1956
AMITSUR 1. Introduction.Let A be an algebra over a field F. We shall be mainly interested in infinite-dimensional algebras over infinite fields F. Actually most of the results will be obtained for nondenumerable fields F.The present paper starts with studying the spectrum of the elements of A. More precisely, the complement of the spectrum in F is studied.It seems that from an algebraic point of view the complement is of greater interest than the spectrum itself.As a consequence of the study of the complement it is shown that the Jacobson radical [2] of algebras A for which (A:F) <the cardinal number of F is always the maximal nil ideal of A. Furthermore, for algebras satisfying this inequality, the fact that an element is algebraic or nil can be expressed by conditions on the complement of the spectrum of that element.These results are then applied to provide an affirmative solution for some well known structure problems of rings, though only for algebras over nondenumerable fields.The first problem is the following problem of Koethe [ó]: "Does the maximal nil ideal of A contain all one-sided nil ideals of A?" The others are two problems proposed by Jacobson [3]: "Let A be an algebraic algebra over F. Does it follow that the matrix ring An over A is also algebraic and is the extension algebra AH algebraic over H for any extension H oí F?"These problems are related to the problem of Kurosh [7] and Levitzki Jo]: "Is every algebraic algebra locally finite?"1It was pointed out by Levitzki [9] that an affirmative solution for this problem will provide an affirmative solution for the problem of Koethe, and it was observed by Jacobson that this will imply also an affirmative answer for the other structure problems of algebraic algebras.In the present paper we attack the problems of Koethe and Jacobson avoiding the problem of Kurosh which still remains unsolved even in the case of nondenumerable fields F, for which we provide an affirmative answer for the other problems.An important tool in attacking the problem of Kurosh are the two radicals defined by Levitzki: the semi-nilpotent radical [8] and the locally finite kernel [lO].The cases of the known affirmative solution