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Semigroup endomorphisms of rings

Abstract We show that rings for which every non-constant multiplicative endomorphism is additive are trivial or power rings (that is, rings R such that R = R2 and x2 = 0 = x+x for all x ∈ R) and that if R is a power ring for which every multiplicative endomorphism is additive, then End (R) is a zero semigroup or a semilattice according to how the product is defined.