Note on Maximal Algebras

Introduction. It has been shown in a previous paper [3]1 that every algebra A with radical R, such that A/R is separable, is a homomorphic image of a certain maximal algebra which is determined to within an isomorphism by A/R, the A/R-modu\e (two-sided) R/R1, 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 B/R is simple. As is well known,' 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 73) is normal over F, in the sense of the noncommutative Galois theory.* This will be the case if and only if the center Z of B/R is a separable normal extension field of F and every automorphism of Z over F is induced by an automorphism of B/R. A completely primary algebra C with radical 5 will henceforth be called quasinormal if C/S is normal over F. If maps the radical FmXS onto the radical R of B, and B/R is isomorphic with FmXC/S. Therefore, if C is quasinormal over F, then B/JZ 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 naturäl extension to B* of a homomorphism of C* onto C is a homomorphism of B* onto 73. From these facts it is evident