a1 Department of Math. and Computer Science, Bar-Ilan University, 52 100 Ramat-Gan, Israel
Sullivan's problem of describing rings, all of whose additive endomorphisms are multiplicative, is generalised to the study of rings R satisfying (a1 … an) = (a1)…(an) for every additive endomorphism of R, and all a1,…,an R, with n > 1 a fixed positive integer. It is shown that such rings possess a bounded (finite) ideal A such that [R/A]n = 0 ([R/A]2n−1 = 0). More generally, if f(X1, …, Xt) is a homogeneous polynomial with integer coefficients, of degree > 1, and if a ring R satisfies [f(a1, …, at)] = f[(a1), …, (at)] for all additive endomorphisms , and all a1, …, at R, then R possesses a bounded ideal A such that R/A satisfies the polynomial identity f.
(Received March 19 1988)