Rings whose additive endomorphisms are multiplicative (Q5905459)

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.