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Rings whose additive endomorphisms are N-multiplicative

Published online by Cambridge University Press:  17 April 2009

Shalom Feigelstock
Shalom Feigelstock
Affiliation:
Department of Math. and Computer Science, Bar-Ilan University, 52 100 Ramat-Gan, Israel
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Abstract

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

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 1 , February 1989 , pp. 11 - 14
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Feigelstock, S., ‘Rings whose additive endomorphisms are multiplicative’ (to appear), in Period. Math. Hungar. to appear.Google Scholar
[2]Fuchs, L., Infinite Abelian Groups I (Academic Press, New York–London, 1971).Google Scholar
[3]Kim, K.H. and Roush, F.W., ‘Additive endomorphisms of rings’, Period. Math. Hungar. 12 (1981), 241–242.CrossRefGoogle Scholar
[4]Sullivan, R.P., ‘Research Problems’, Period. Math. Hungar 8 (1977), 313–314.Google Scholar