Some remarks on \(G\)-Noetherian rings (Q2785910)

Let \(R\) be a commutative ring with identity. A prime ideal \(P\) of \(R\) is called a \(G\)-ideal if \(P=M\cap R\) for some maximal ideal \(M\) of the polynomial ring \(R[X]\). The ring \(R\) is called \(G\)-Noetherian if it satisfies the ascending chain condition on \(G\)-ideals. The ring \(R\) is called a Hilbert ring if every \(G\)-ideal is maximal. It is proved that the following statements are equivalent for a field \(K\) and a set \(\{X_\lambda\}_{\lambda\in\Lambda}\) of indeterminates over \(K\): NEWLINENEWLINENEWLINE(i) the polynomial ring \(K[\{X_\lambda\}]\) is a Hilbert ring, NEWLINENEWLINENEWLINE(ii) \(|\Lambda|<|K|\aleph_0\), NEWLINENEWLINENEWLINE(iii) \(K[\{X_\lambda\}]\) is \(G\)-Noetherian. NEWLINENEWLINENEWLINEThe authors comment on two papers of \textit{R. Rajagopalan} [namely ``\(G\)-Noetherian domains'', Indian J. Pure Appl. Math. 21, No. 10, 919-921 (1990; Zbl 0724.13016)] and ``On \(G\)-ideals and \(G\)-Noetherian domains'', Simon Stevin 67, Suppl., 59-63 (1993; Zbl 0833.13006)] extending some of the results and giving various corrections.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].