Inner Derivations of Division Rings and Canonical Jordan form of Triangular Operators

Abstract

Let D be a division ring and k its center. We show that a generalized canonical Jordan form exists for triangularizable matrices A over D which are algebraic over k, i.e, satisfy f(A) = 0 for some nonzero polynomial f over k. This canonical form is a direct sum of generalized Jordan blocks Jrn (a, /3). This block is an m by m matrix whose diagonal entries are equal to a, those on the first superdiagonal are equal to /3, and all other entries are equal to zero. If a is separable over k then we can choose /3 = 1, but in general this cannot be done. Notation. D denotes a division ring, and k its center. By X we denote a fixed element of D, and by K the centralizer of X in D. We set D* = D {0}, K * = K {0} and U = { Xa aX: a E D }. The right D-vector space whose elements are column n-vectors over D is denoted by D n and its standard basis is denoted as usual by { el, e2, . . ,en }. By R = D[t] we denote the ordinary polynomial ring over D in the indeterminate t. If f E k[t] then (f ) will denote the ideal of k[t] generated by f while fR ( = Rf ) will denote the ideal of R generated by f. Following [7] and [3] we shall say that an R-module is bounded if its annihilator (also called its bound) is not the zero ideal. For a, /8 E D we denote by Dn(a, /3) the vector space Dn considered as a right R-module in which t acts as a D-linear transformation such that elt = ela, and eit = ei a + ei_-/3 for 1 < i < n. Clearly, if /8 # 0 this module is cyclic with generator en. The 1-dimensional module Dl(a, ,8) is independent of /3 and will be denoted by D(a). The module Dn(a, ,8) has length n and each factor of its Jordan-Holder series is isomorphic to D(a). Mn(D) is the k-algebra of n by n matrices over D. X Y for X, Y E Mn(D) means that X is similar to Y. Let Jm(a, 8) E Mm(D) denote the generalized Jordan block with diagonal entries equal to a and those on the first superdiagonal equal to /3, while all other entries are zeros. We shall write Jm((a) for the usual Jordan block Jm(a, 1). For a E D we define [a] to be the set { 8a-y: /3, y E K }. Note that, in general, [a] is not closed under addition. Received by the editors November 16, 1983 and, in revised form, February 22, 1984 and September 10, 19F4. 1YSO Mathematics Subject Classification. Primary 16A42, 16A72; Secondary 12E15, 16A39. Kev words and phrases. Annihilator, Jordan-Holder series, generalized Jordan blocks, irreducible polynomial, bounded module, similarity. 'Research supported by NSERC Grant A-5285. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page