Rings virtually satisfying a polynomial identity

Abstract

Let R be a ring and f(x1,..., xn) be a polynomial in noncommutative indeterminates x1,..., xn with coefficients from ℤ and zero constant. The ring R is said to be an f-ring if f(r1,..., rn) = 0 for all r1,..., rn of R and a virtually f-ring if for every n infinite subsets X1,..., Xn (not necessarily distinct) of R, there exist n elements r1 ∈ X1,...,rn ∈ Xn such that f(r1,..., rn) = 0. Let R* be the 'smallest' ring (in some sense) with identity containing R such that Char (R) = Char (R*). Then denote by ZR the subring generated by the identity of R* and denote by f̄R the image of f in ZR [x1,..., xn] (the ring of polynomials with coefficients in ZR in commutative indeterminates x1,..., xn). In this paper, we show that if R is a left primitive virtually f-ring such that f̄R ≠ 0, then R is finite. Using this result, we prove that an infinite semisimple virtually f-ring R is an f-ring, if the subring of ZR generated by the coefficients of f̄R is equal to ZR; and we also prove that if f(x) = ∑i=2n aixi + εx ∈ ℤ [x], where ε ∈ {-1, 1}, then every infinite virtually f-ring with identity is a commutative f-ring. Finally we show that a commutative Noetherian virtually f-ring R with identity is finite if the subring generated by the coefficients of f̄R is ZR. © 2004 Elsevier B.V. All rights reserved.