Bulletin of the Australian Mathematical Society

Research Article

Rings whose additive endomorphisms are N-multiplicative

Shalom Feigelstocka1

a1 Department of Math. and Computer Science, Bar-Ilan University, 52 100 Ramat-Gan, Israel

Abstract

Sullivan's problem of describing rings, all of whose additive endomorphisms are multiplicative, is generalised to the study of rings R satisfying xs03D5(a1an) = xs03D5(a1)…xs03D5(an) for every additive endomorphism xs03D5 of R, and all a1,…,an xs2208 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 xs03D5[f(a1, …, at)] = f[xs03D5(a1), …, xs03D5(at)] for all additive endomorphisms xs03D5, and all a1, …, at xs2208 R, then R possesses a bounded ideal A such that R/A satisfies the polynomial identity f.

(Received March 19 1988)