General polynomial identities. II

A generalized polynomial is a formal polynomial with coefficients in a noncommutative ring (so that the coefficients are interspersed with the indeterminates); a generalized monomial of a generalized polynomial is a sum of all monomials in which the indeterminates occur in the same order. A generalized polynomial identity (GI) of R is R-proper if one of its generalized monomials is not a GI of R. The two main results of this paper are: (i) Any GI of a primitive ring P is P socP-improper, where soc P is the socle of P; (ii) if R satisfies a GI that is proper for every nonzero homomorphic image of R, then R is a PI-algebra (in the usual sense). These theorems have several immediate applications: 1. (i) If some element r in R is a root of an algebraic equation f(λ) = 0 with f′(r) invertible, f′ the formal derivative of f, and if the centralizer of r is a PI-algebra, then R is a PI-algebra (most of this was already known by M. Smith); 2. (ii) If R has two left ideals A and B that are PI-algebras, then A + B is a PI-algebra. The main results of the paper are also given in the context of rings with involution.