Rings virtually satisfying a polynomial identity

Let R be a ring and f(x1,…,xn) be a polynomial in noncommutative indeterminates x1,…,xn with coefficients from image 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 r1set membership, variantX1,…,rnset membership, variantXn 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 image 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 image, 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 image is equal to ZR; and we also prove that if image, where εset membership, variant{-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 image is ZR.