Strongly filial rings (Q330648)

The author introduces and studies strongly filial rings. A ring \(R\) is \textit{strongly filial} if for every \(a\in R,\) \((a)_{R}=(a^{2})_{R}+\mathbb{Z}a\). The weaker notion called filial rings have already been studied extensively. This stronger form also captures when certain subrings are actually ideals. In particular, it is shown that \(R\) is strongly filial if and only if every subring of \(R\) which is both \(n\)-left accessible and \(n\)-right accessible in \(R\) for some \(n\geq 1\) is an ideal of \(R\). Various other characterizations of these rings are given and it is shown that the prime radical of a strongly filial ring coincides with the sum of all the nilpotent ideals as well as with the set of all nilpotent elements. Strongly \(f\)-regular rings have been defined earlier as those rings for which all homomorphic images are reduced. The relationship between strongly filial rings and strongly \(f\)-regular rings is also discussed.