Localization and ideal theory in Noetherian strongly group-graded rings
A right Noetherian ring R satisfies the right second layer condition if, for every prime ideal P of R, whenever M is a finitely generated uniform right R-module which contains a copy of a non-zero right ideal of R/P and has prime annihilator Q, we have \(P=Q\). This condition is crucial to the theory of localization at (possibly infinite) sets of prime ideals of R and to the representation theory of R. It is known to hold in many (though not all) Noetherian rings. Details can be found in [\textit{A. V. Jategaonkar}, Localization in Noetherian rings (Lond. Math. Soc. Lect. Note Ser. 98, 1986; Zbl 0589.16014)]. The main result of this paper (7.4), states that if G is a polycyclic-by- finite group and S is a strongly G-graded ring with right Noetherian base ring R, then S has the right second layer condition provided R is right Artinian, or G-simple, or satisfies a polynomial identity. This partially generalizes earlier work of Jategaonkar and the reviewer on ordinary group rings - consult the above reference. In developing the background results needed to prove this result, the author provides an overview of much of what is known about rings strongly graded by a polycyclic-by- finite group, and obtains several interesting subsidiary results.
finitely generated uniform right R-module
scientific article; zbMATH DE number 3981376
scientific article