N-rings and ACC on colon ideals
\textit{R. Gilmer} and \textit{W. Heinzer} [cf. ibid. 24, 123-144 (1982; Zbl 0495.13002)] introduced the notion of N-ring, that is, a ring R such that, for every ideal A of R, there exists a Noetherian ring extension S of R such that \(AS\cap R=A\). Although an N-ring need not be Noetherian (an example attributed to M. Hochster, of a one-dimensional quasilocal non- Noetherian integral domain which is an N-ring is given here), in this paper the authors show that such a ring shares many of the usual properties of a Noetherian ring. For instance, every finite integral extension of an N-ring is an N-ring. This result is a consequence of one of the main theorems of this paper: A ring R is an N-ring if and only if, for every ideal A of R, the ring R/A satisfies the ascending chain condition on annihilator ideals.
ascending chain condition on annihilator ideals
scientific article; zbMATH DE number 3922817