A note on additive subgroups of finite rings (Q1841828)

Let \(R\) be a ring with identity. A subring \(S\) is said to be a subideal if there exists an increasing finite sequence of subrings such that the first one is equal to \(S\), the last to \(R\) and every of them is an ideal in the following one. The main result of the article is the following theorem: If \(R\) is a finite ring with identity, \(S\) is a subgroup of its additive group and either \(R\) is semisimple or \(S\) is a subring of \(R\), then the following conditions are equivalent: \(S\) is a subideal in \(R\); \(S\) is a subideal of the ring generated by \(S\) and \(r\) for every \(r\in R\); \(S\cap T\) is a subideal in \(T\) for every \(2\)-generator subring \(T\); for every \(s\in S\) and for every \(r\in R\) there exists \(n\in\mathbb{N}\) such that both \((sr)^n\in S\) and \(r(sr)^n\in S\).