Rings virtually satisfying a polynomial identity

Abstract: Let R be a ring and be a polynomial in noncommutative indeterminates with coefficients from and zero constant. The ring R is said to be an f-ring if for all of R and a virtually f-ring if for every n infinite subsets (not necessarily distinct) of R, there exist n elements such that . Let be the ‘smallest’ ring (in some sense) with identity containing R such that . Then denote by the subring generated by the identity of and denote by the image of f in (the ring of polynomials with coefficients in in commutative indeterminates ). In this paper, we show that if R is a left primitive virtually f-ring such that , 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 generated by the coefficients of is equal to ; and we also prove that if , where , 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 is . [Copyright &y& Elsevier]