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Rings whose additive endomorphisms are multiplicative. (English) Zbl 0754.16001

Let \(R\) be an \(AE\)-ring (i.e., every endomorphism of the additive group \(R(+)\) is also an endomorphism of the multiplicative semigroup \(R(\cdot)\)). Then \(R\) is said to be non-trivial if \(R^ 2\neq 0\) and \(R\) is said to be standard if \(R\) is the ring direct sum \(R=Q\oplus S\oplus T\) where \(Q(+)=\langle q\rangle\) is a cyclic group of order \(2^ n\), \(n\geq 1\), \(q^ 2=2^{n-1}q\), \(S^ 2=2^{n-1}S=0\), \(2T=T\) and \(T^ 2=T_ 2=0\) (here, \(T_ 2\) denotes the subgroup of \(T(+)\) consisting of all elements of 2-power order). If \(R\) is non-trivial, then the following conditions are equivalent: (1) \(R\) is standard; (2) \(R^ 2\nsubseteq\cup 2^ kR\); (3) \(\cup 2^ kR_ 2=0\); (4) \(R_ 2\) is bounded; (5) \(R_ 2\) is a non-trivial \(AE\)-ring; (6) The torsion part \(R_ t\) of \(R(+)\) is a non-trivial \(AE\)-ring. If \(R\) is not standard, then \(R(R_ t+2R)=0=(R_ t+2R)R\). The paper is concluded with a nice example of a non-trivial and non-standard \(AE\)-ring.
Reviewer: T.Kepka (Praha)

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16W20 Automorphisms and endomorphisms
20K21 Mixed groups
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References:

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