Finite Dimensional Central Simple Algebra Research Articles

*-Generalized polynomial identities of finite dimensional central simple algebras

LetD be a finite dimensional central simple algebra with involution * of the first kind. We prove (in a fixed number of variables) the ideal of *-GPI’s ofD is generated by a finite collection of elements of the form [v ij ,v pq ] andv ij −v ij * where thev ij ’s are first degree generalized polynomials.

Azumaya algebras with involution

One of the highpoints of the theory of central simple algebras as developed in the 1920s and 1930s was the results of Albert concerning simple rings with involution. A part of his results was the characterization of those finite dimensional central simple algebras which admit an involution. This characterization was extended and clarified by Scharlau [8] and Tamagawa (unpublished). It is our intention here to generalize this result in the context of Azumaya algebras over commutative rings. We obtain conditions parallel to the classical ones as to when the equivalence class of an Azumaya algebra contains an algebra with involution. In the last section, we improve and clarify our result in three special cases; Azumaya algebras of rank 4, trivial Azumaya algebras, and Azumaya algebras over connected semilocal rings. Contained in the arguments of this paper is a new proof of the classical result.

Numerical invariants and projective modules

In a recent paper [9], Fr6hlich gives criteria in terms of numerical invariants to insure the projectivity of modules of an order in a commutative separable algebra over the quotient field of a Dedekind domain. More precisely, Frijhlich first defines a numerical invariant, the module index (see [8] for the definition), and then considers the following context: let A be a Dedekind domain, K its quotient field, 2 a finite dimensional commutative separable K-algebra, r the unique maximal A-order in Z and A any other A-order of Z. Now if M is a A-module such that M @A K E LY’), for an integer I, then M is A-projective if and only if [FM : M] = [r : A]’ (I’M is taken to denote the smallest r-module containing M, and [P : Q] is the notation for the module index of A-lattices P and Q). For purposes of reference we call this result “Friihlich’s theorem.” The object of this paper is to extend and give analogues to FrBhlich’s results for the case that Z is a finite dimensional separable K-algebra. It will be seen, however, that in the fairly simple case of a finite dimensional matrix algebra over the quotient field of a (complete) discrete rank one valuation rank, no direct analogue of either the necessity or the sufficiency of Fr6hlich’s theorem can be given. Nevertheless, for any finite dimensional central simple algebra Z over the quotient field of a complete discrete rank one valuation ring, we can give results similar to those of Frijhlich for some special types of A-orders in 2. Further, we shall show that Fr6hlich’s results extend almost entirely when Z is a central division K-algebra and the base ring A is a complete discrete rank one valuation ring. FrGhlich also gave a criteria to determine the projectivity of fractional ideals I in A such that I BA K E 2. This can be stated as: I is A-projective

The Freudenthal-Springer-Tits constructions revisited

In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

Subrings of simple rings with minimal ideals

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 Dieudonne [1 ] 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 9, T over a division ring D. 9 and T are linked by an inner product (x, f), x in 9, f in W, which is a nondegenerate bilinear function from 9 X T to D. A linear transformation (l.t.) a on 9 is said to have an adjoint a* on T if there is a l.t. a* on T 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 9a is finite-dimensional. It may then be shown that any continuous finite-valued linear transformation (f.v.l.t.) a on 9) has the form za = Z:(z, fi)xi, where z, xi are in 9, and thefi are in 9. The ring A may then be regarded as the ring 7(A9, %) of all f.v.l.t. on 9) which have adjoints on 9. We shall use the notation Q(j2, %) for the ring of all continuous l.t. on W. Furthermore A =7(9, %) is a dense ring of f.v.l.t. on 9W. That is, for any n linearly independent vectors x1, x2, * x, of T and any n arbitrary vectors yi, Y2, * * *, yn of 9, there is a l.t. a in A such that xia =yi. Conversely any dense ring of f.v.l.t. on 9 is an S.M.I. ring. If 9 is a subspace of 91, the annihilator of S in 9 is the set of all vectors f of T such that (S, f) =0. The annihilator is a subspace of 9 and will be