Generalized local rings and finite generation of powers of ideals (Q2752912)

This paper consists of two parts. The first part reviews conditions in order that a generalized local ring (that is a quasilocal ring \((R,M)\) in which \(M\) is finitely generated and \(\bigcap_{n \geq 1}M^n = 0\)) be Noetherian and presents several examples of non-Noetherian generalized local rings. In particular, it establishes the existence of a quasilocal domain \((R,M)\) such that both the ideal \(M\) and the ideal \(\bigcap_{n \geq 1}M^n\) are finitely generated and yet \(\bigcap_{n \geq 1}M^n \neq 0\). NEWLINENEWLINENEWLINEThe second part of the paper addresses the problem whether the maximal ideal \(M\) of an integral domain \(R\) is finitely generated if some power of \(I\) is finitely generated. In general, this problem has a negative answer. The paper discusses in detail recent affirmative results proved by \textit{R. Gilmer, W. Heinzer} and \textit{M. Roitman} [Proc. Am. Math. Soc. 127, 3141-3151 (1999; Zbl 0923.13012)] and \textit{M. Roitman} [J. Pure Appl. Algebra 161, No. 3, 327-340 (2001; Zbl 1034.13010)].NEWLINENEWLINEFor the entire collection see [Zbl 0964.00012].