A visit to maximal non-ACCP subrings (Q2788761)

\textit{A. Ayache} et al. in [J. Algebra Appl. 6, No. 5, 873--894 (2007; Zbl 1154.13007)] considered integral domains \(R\) with quotient field \(K\) such that \(R\) is maximal with respect to having quotient field \(K\) and not satisfying the ACCP (i.e., the ascending chain condition on principal ideals). They gave some characterizations of ``maximal non-ACCP subrings'' in specific cases, such as the case where \((R,S)\) is a residually algebraic pair (i.e., every intermediate ring \(T\), \(R\subseteq T \subseteq S\), is residually algebraic) and \(R\) is local. In this paper, the authors extend the previous work and characterize maximal non-ACCP subrings \(R\) of an integral domain domain \(S\) in case \((R, S)\) is a residually algebraic pair and R is semilocal.NEWLINENEWLINEFurthermore, given a field \(K\), they also consider a \(K\)-algebra \(S\), a nonzero proper ideal \(I\) of \(S\) and a subring \(D\) of the field \(K\) and they determine necessary and sufficient conditions in order that the ring \(D + I\) is a maximal non-ACCP subring of \(S\). As an application, they give an example of a maximal non-ACCP subring \(R \) of a domain \(S\) such that \((R, S)\) is a normal pair (i.e., every intermediate ring \(T\), \(R\subseteq T \subseteq S\), is integrally closed in \(S\)) and \( R \) is not semilocal.