An independence property of central polynomials

Chen-Lian Chuang

Vol. 148 (1991), No. 2, 237–249

DOI: 10.2140/pjm.1991.148.237

Abstract

Let Φ_n be the ring of n × n matrices over a commutative field Φ. Let f_i(x₁,…,x_m) and g_i(y₁,…,y_m) (i = 1,…,k) be polynomials with coefficients in Φ and with noncommuting indeterminates in the disjoint sets {x₁,…,x_m} and {y₁,…,y_m}. Assume that f₁(x₁,…,x_m),…,f_k(x₁,…,x_m) are Φ-independent modulo the T-ideal of polynomial identities of Φ_n. Consider the following two statements: (1) whenever ∑ _i=1^kf_i(x₁,…,x_m)g_i(y₁,…,y_m) is central on Φ_n, then so is each g_i(y₁,…,y_m) (i = 1,…,k); (2) whenever ∑ _i=1^kf_i(x₁,…,x_m)g_i(y₁,…,y_m) is a polynomial identity for Φ_n, then so is each g_i(y₁,…,y_m) (i = 1,…,k). It is shown here that statement (2) is always true and that statement (1) holds but for the exceptional case: n = 2 and Φ is the ring of integers modulo 2.

Mathematical Subject Classification 2000

Primary: 16R10

Secondary: 16R99

Milestones

Received: 15 August 1989

Revised: 1 March 1990

Published: 1 April 1991