Chain conditions in special pullbacks (Q456621)

In the paper under review, the authors study the ascending chain condition on the ideals (Noetherian poperty) and on the principal ideals (ACCP) of extensions of commutative rings via monoids.NEWLINENEWLINEMore precisely, let \(D\) be a commutative ring with identity, \(I\) an ideal of \(D\), \(E\supseteq D\) another commutative ring and \(\Gamma\) a non-zero torsion-free additive grading monoid with \(\Gamma \cap -\Gamma=\{0\}\). Set \(\Gamma^*=\Gamma\setminus \{0\}\).NEWLINENEWLINEIt is shown in Theorem 2.1 (resp. 2.4) that \(D+E[\Gamma^*]\) (resp. \(D+I[\Gamma^*]\)) is Noetherian if and only if \(D\) is Noetherian, \(E\) is finitely generated as a \(D\)-module (resp. \(I=I^2\)) and \(\Gamma\) is finitely generated. Theorem 2.1 is a generalization of a result of \textit{S. Hizem} [Commutative algebra and its applications. Proceedings of the fifth international Fez conference on commutative algebra and applications, Fez, Morocco, June 23--28, 2009. Berlin: Walter de Gruyter. 259--274 (2009; Zbl 1177.13044)].NEWLINENEWLINEAbout the ACCP condition, it is shown in Theorem 3.4 (resp. 3.7) that if \(D\) is an integral domain, then \(D+E[\Gamma^*]\) (resp. \(D+I[\Gamma^*]\)) satisfies ACCP if and only if \(\Gamma\) satisfies ACCP and \(\bigcap_{n\geq 1}a_1\cdots a_n E=0\) for every sequence \((a_n)\) of non-unit elements of \(D\) (resp. \(\Gamma \) and \(D\) satisfy ACCP).