On radicals of ring extensions
\textit{B. de la Rosa} and \textit{G. A. P. Heyman} have shown that a ring radical R satisfies \(R(A)=R(A^*)\), for every Dorroh extension \(A^*\) of a ring A, if and only if \(R({\mathbb{Z}})=0\), \({\mathbb{Z}}\) the ring of integers [Arch. Math. 42, 516 (1984; Zbl 0522.16008)]. The author of this paper proves the following fact. Theorem. Let R be a ring radical, and let B be a ring. The following are equivalent: 1) There exists an extension \(R(B)^*\) of R(B) by B such that \(R[R(B)^*]=R(B)\). 2) \(R(B)=0\). 3) For every extension \(A^*\) of a ring A by B, \(R(A)=R(A^*)\).
scientific article; zbMATH DE number 3981368