RINGS WHOSE ADDITIVE ENDOMORPHISMS ARE RING ENDOMORPHISMS

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed — the last open case — by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed. © 1990, Australian Mathematical Society. All rights reserved.