Rings virtually satisfying a polynomial identity

Abstract

Let R be a ring and f (x 1 , . . . , x n ) be a polynomial in noncommutative indeterminates x 1 , . . . , x n with coefficients from Z and zero constant. The ring R is said to be an f-ring if f (r 1 , . . . , r n ) = 0 for all r 1 , . . . , r n of R and a virtually f-ring if for every n infinite subsets X 1 , . . . , X n (not necessarily distinct) of R, there exist n elements r 1 ∈ X 1 , . . . , r n ∈ X n such that f (r 1 , . . . , r n ) = 0. Let R * be the 'smallest' ring (in some sense) with identity containing R such that Char(R) = Char(R * ). Then denote by Z R the subring generated by the identity of R * and denote byf R the image of f in Z R [x 1 , . . . , x n ] (the ring of polynomials with coefficients in Z R in commutative indeterminates x 1 , . . . , x n ). In this paper, we show that if R is a left primitive virtually f-ring such thatf R = 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 Z R generated by the coefficients off R is equal to Z R ; and we also prove that if f (where ∈ {−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 off R is Z R .