Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions

Abstract

Let k denote an algebraically closed field of arbitrary characteristic. Let C denote the set of all commutative, finite dimensional, local k-algebras of the form (B, m, k) with i(m) ⋚2. Here i(m) denotes the index of nilpotency of the maximal ideal m. A Akalgebra (R, J,k)∈L is called a (c1-construction if there exists (B, m, k)∈ £ ≅ {(k, (0), k)} and a finitely generated, faithful B-module N such that R,≅B⋉(the idealization of N). (R.J.k) is called a (c2::-construction if there exist a (B,m k)∈ L, a positive integer p $ge;2 and a nonzero z £ SB(the socle of B) such that R≅B[x]/(mX, Xp- z). Let Mn×n(K) denote the set of all n x n matrices, over k with n≥2. Let .Mn(k) denote the set of all maximal, commutative A;-subalgebras of Mn×n(k). In this paper, we show any (R J, k) ∈£⋂Mn;(k) with n>5 is a C1 or C2 -construction except for one isomorphism class. The one exception occurs when n = 5.