Maximal subalgebras of central separable algebras

Abstract

Abstract. Let A be a central separable algebra over a commutative ring R. A proper Ä-subalgebra of A is said to be maximal if it is maximal with respect to inclusion.Theorem. Any proper subalgebra of A is contained in a maximal one. Any maximal subalgebra B of A contains a maximal ideal mA of A, m a maximal ideal of R, and B/mA is a maximal subalgebra of the central simple R/m algebra A/mA.More intrinsic characterizations are obtained when ii is a Dedekind domain.In [3] and [4] Dynkin studied maximal subalgebras of simple Lie algebras over an algebraically closed field of characteristic zero and the connection between these and the maximal subgroups of the classical groups. In [7] and [8] the maximal subalgebras of the following classes of central simple algebras were determined: associative, associative with involution, alternative and Jordan. In this paper maximal subalgebras of central separable algebras over a commutative ring R are considered. The ideal structure of such algebras is entirely determined by that of R so it is interesting to consider one-sided ideals and to a lesser extent subalgebras.We recall some results of [7]. Let F be a field and A a (finite dimensional) central simple algebra over F. So A se Mn(D), D a finite dimensional division algebra over its center F and A acts on an «-dimensional left D vector space V. By a subalgebra of A we understand an F subspace of A which is closed under multiplication, and by maximal subalgebra, a proper subalgebra which is maximal with respect to inclusion. If A = F then 0 is the unique maximal subalgebra of A. If A =£ F then all maximal subalgebras of A contain 1 and they are exactly the subalgebras of the form:(I)S(WO = {a E A\Wa c W), W any nonzero proper subspace of V.

Maximal Subrings and Covering Numbers of Finite Semisimple Rings

“…We return now to the situation of a central simple algebra A with center F . In [21], Racine proved that maximal F -subalgebras of A fall into two types (Racine later gave an analogous classification of maximal subalgebras of central separable algebras over a commutative ring R in [22]). Note that if R is an F -subalgebra of A containing 1 A , S is a subring of A , and R ⊆ S, then F ⊆ S, so that S is also an F -subalgebra.…”

“…7] and [5] relied on a classification of the maximal subrings of Mn(Fq) that did not include Type III maximal subrings (see [17,Lem. 7.1], which is equivalent to Racine's classifications [21,22] of maximal Fq-subalgebras of Mn(Fq)). However, the techniques employed in [17] and [5] were robust enough that Type III maximal subrings where automatically excluded from the minimal covers that were formed, and so σ(Mn(Fq)) can be determined by using only Type I and Type II maximal subrings.…”

“…There are many papers in the literature studying maximal subalgebras of an algebra A over a field F (see e.g. [9,10,11,21,22,23] and the references therein), and for commutative rings the related notion of minimal ring extensions has become popular (for this, the papers [6,7,8] are a good starting point). However, less work has been done on classifying the maximal subrings of a general algebra A .…”

“…The paper begins in Section 2 with some results on maximal subrings of a finite dimensional central simple algebra A over a field F . This allows us to build on the work of Racine [21,22], who classified maximal F -subalgebras in various types of F -algebras, among them matrix algebras and central simple algebras. As we demonstrate in Proposition 2.4 and Theorem 3.3, as long as Fq = Fp it is easy to produce maximal subrings of Mn(Fq) that are not Fq-subalgebras.…”

Automorphism Groups of Finite-Dimensional Algebras Acting on Subalgebra Varieties

“…Maximal commutative subalgebras of M n (k) have been studied extensively, for instance in [37], [23], [27], [28], [29], [18], [25]. Maximal subalgebras of central simple algebras were classified by Racine in [33], [34]. Independently, Agore [1] used a geometric argument of Gerstenhaber [17] to bound the maximal dimension of a C-subalgebra in M n (C) by n 2 −n+1.…”