JOURNAL OF ALGEBRA 23, 129-l 32 (1972) Central Polynomials for Matrix Rings EDWARD FORMANEK* Department of Mathematics, Carleton University, Ottawa, Ontario, Canada Communicated by I. N. Herstein Received October 15, 1971 DEDICATED TO DEBORAH TEPPER HAIMO IN HONOR OF HER FIFTY-FIRST BIRTHDAY, JULY 1, 1972 Let K be a field, A an K-algebra, and R = K[X, ,..., X,] the polynomial ring over K in noncommuting variables XI ,..., X, . A polynomial identity for A is a polynomial F(X, ,..., X,) E R such that F(a, ,..., a,) = 0 for any a, ,..., a, E A; a central polynomial for A is a polynomial F in R such that F(a, ,..., a,) E center A for any a, ,..., a, E A; a central polynomial is non- vanishing if it is not a polynomial identity. The existence of nonvanishing central polynomials for matrix rings over a field was one of the problemsgiven by Kaplansky in [I]. Central polynomials for matrix rings over finite fields were given in [2]; for infinite fields only the central polynomial (XY - YX)2 for 2 x 2 matriceswashitherto known. We will construct a family of central polynomials, one for each 71, which, like the polynomial (XY - YX)2, are “universal” central polynomials in that they have integer coefficientsand arenonvanishing on M,(A) if A is a commutative ring with unit. THEOREM. If K is a $eld, Mn(K) has a nonvanishing central polynomial. Proof. Let x1 ,..., x,+~ be commuting variables, and X, YI ,..., Y, non- commuting variables. The function associates a polynomial in K[X, YI ,..., Y,] to each polynomial in K[x, ,..., x,+J. Let G(X, YI ,..., Y,) be the polynomial correspondingto * The author is a Canadian NRC postgraduate fellow supported by Grant A7171. 129 Copyright 0 1972 by Academic Press, Inc. All rights of reproduction in any form reserved.