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SUBRINGS OF SIMPLE RINGS WITH MINIMAL IDEALS(l)
| Content Provider | Semantic Scholar |
|---|---|
| Copyright Year | 2010 |
| Abstract | 0. Introduction. In the classical theory of finite-dimensional central simple algebras a large part of the work deals with a study of the simple subalgebras of such an algebra. In particular, the commutators of simple subalgebras as well as pairs of isomorphic subalgebras have been studied by A. A. Albert and E. Noether. For a modern presentation of their principal results we refer to [3, pp. 101-104](2). It is the purpose of this paper to study extensions of these results to simple rings which possess a minimal onesided ideal (S.M.I, rings). Throughout the work we shall use the words simple ring to mean a ring which has no radical and no proper two-sided ideals. It should be noted that it is not known whether a non-nilpotent ring with no proper two-sided ideals may be a radical ring or not. The words radical and semi-simple will be used in the sense of Jacobson [4]. The structure theory of S.M.I, rings has been given by Dieudonné [l ] and Jacobson [5], and will be briefly summarized here. With every S.M.I, ring A there is associated a pair of dual vector spaces 93?, 9? over a division ring D. 93? and 9? are linked by an inner product (x, /), x in 93?, / in 9?, which is a nondegenerate bilinear function from 93? X 9? to P. A linear transformation (l.t.) a on 93? is said to have an adjoint a* on 9? if there is a l.t. a* on 9? such that (xa, f)=(x, a*f). The l.t. which possess adjoints are often called continuous. We shall call a l.t. a finite-valued if the space 93?a is finite-dimensional. It may then be shown that any continuous finite-valued linear transformation (f.v.l.t.) a on 93? has the form za = ^¿(z,/,)*», where z, x¿ are in 93?, and the/,are in 9?. The ring A may then be regarded as the ring 7(93?, 91) of all f.v.l.t. on 93? which have adjoints on 9?. We shall use the notation .£(93?, 9?) for the ring of all continuous l.t. on 93?. Furthermore A =7(93?, 9?) is a dense ring of f.v.l.t. on 93?. That is, for any n linearly independent vectors X\, x2, • • •, xn of 93? and any n arbitrary vectors yi, y2, • • ■ , yn of 93?, there is a l.t. a in A such that x,a=y¿. Conversely any dense ring of f.v.l.t. on 93? is an S.M.I, ring. If $ is a subspace of 93?, the annihilator of $ in 9? is the set of all vectors / of 9? such that ($, /) =0. The annihilator is a subspace of 9? and will be |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/tran/1952-073-01/S0002-9947-1952-0049163-5/S0002-9947-1952-0049163-5.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
