1. Note on maximal algebras
- Author
-
G. Hochschild
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Normal extension ,Division algebra ,Field (mathematics) ,Homomorphism ,Center (group theory) ,Isomorphism ,Automorphism ,Noncommutative geometry ,Mathematics - Abstract
Introduction. It has been shown in a previous paper [311 that every algebra A with radical R, such that AIR is separable, is a homomorphic image of a certain maximal algebra which is determined to within an isomorphism by A/R, the A /R-module (two-sided) R/R2, and the index of nilpotency of R. Furthermore, some indication was given of how the structure of maximal algebras can be determined in simple cases. Here, we wish to give a further illustration by describing a rather wide class of maximal and primary algebras whose structure will be shown to resemble that of crossed products, in certain respects. In fact, we shall impose a certain normality condition and then trace the consequences of a few simple facts of the noncommutative Galois theory. An algebra B over the field F, with radical R, is called primary if it has an identity element and if BIR is simple. As is well known,2 B is then isomorphic with a Kronecker product FmXC, where Fm denotes the full matrix algebra of degree m over F, and where C is completely primary, in the sense that it has an identity element and that the quotient of C by its radical is a division algebra over F. We are concerned with primary algebras B for which this division algebra (which is determined to within an isomorphism by B) is normal over F, in the sense of the noncommutative Galois theory.3 This will be the case if and only if the center Z of BIR is a separable normal extension field of F and every automorphism of Z over F is induced by an automorphism of BIR. A completely primary algebra C with radical S will henceforth be called quasinormal if C/S is normal over F. If 4 is an isomorphism of Fm X C onto B then 4 maps the radical Fm XS onto the radical R of B, and B/R is isomorphic with Fm X C/S. Therefore, if C is quasinormal over F, then B/;R is automatically separable over F. By [3], B has then a maximal related extension B* which is evidently primary. Moreover, it is easily seen that B* FmXC*, where C* is the maximal related extension of C, and is quasinormal over F. Finally, the natural extension to B* of a homomorphism of C* onto C is a homomorphism of B* onto B. From these facts it is evident
- Published
- 1950