\(\Omega\)-Krull rings and Gr-\(\Omega\)-Krull rings in the non-P.I. case
The class of \(\Omega\)-Krull rings consists of noncommutative rings possessing properties generalizing those of Krull domains. The prefix \(\Omega\) refers to \(\Omega\)-rings which had been mentioned earlier by E. Nauwelaerts, F. Van Oystaeyen and \(\Omega\) refers to the fact that these rings are characterized by conditions on the maximal ideals. One may also give an intrinsic graded definition when the rings considered are graded, mostly by a torsion free abelian group or cancellative monoid here, and doing this one obtains the class of Gr-\(\Omega\)-Krull rings. The main result of this note provides sufficient and necessary conditions for a graded ring R to be a central \(\Omega\)-Krull ring in terms of the graded property for R and the ungraded property of the graded symmetric ring of quotients of R. The author applies this result to the study of certain special generalized Rees rings and in particular he derives conditions for a (twisted) group ring over a torsion free abelian group to be an \(\Omega\)-Krull ring. In the P.I. case \(\Omega\)-Krull rings will be just Krull orders and then other and more specific results are known. The use of the symmetric ring of quotients allows to obtain results in the non-PI case and this includes some examples over simple Noetherian non-Artinian rings.
scientific article; zbMATH DE number 3859328