Über N. I. Dubrovin's Ansatz zur Konstruktion von nicht vollprimen Primidealen in Kettenringen. (On N. I. Dubrovin's attempt to the construction of not completely prime prime ideals in chain rings)
A chain ring R is a ring with identity for which aR\(\subseteq bR\) or bR\(\subseteq aR\) holds for any a,b in R and the corresponding condition for left ideals. These rings appear in geometry, module theory and as noncommutative valuation rings in division algebras. The present article begins with a review of papers of the past twenty-five years in which the following question was encountered: Does there exist a chain ring R with a prime ideal P that is not completely prime; i.e. AB\(\subseteq P\), \(A\varsubsetneq P\) implies \(B\subseteq P\) for ideals A,B of R, but there exist elements a,b in R with ab\(\in P\) and both a,b\(\not\in P\). The question remains open and it is the purpose of this paper to point out that the gap in the construction of an example for such a ring as suggested by \textit{N. I. Dubrovin} [Math. USSR, Sb. 48, 437-444 (1984), in particular: Lemma 3, p. 439; translation from Mat. Sb., Nov. Ser. 120, 441-447 (1983; Zbl 0517.16002)] cannot be easily filled. The ring in that example was constructed as a localization (in the sense of P. M. Cohn) of some subring of a certain group ring. Here it is shown by a rather ingenious example that a crucial property is not preserved under localization as was assumed in the above mentioned paper.
noncommutative valuation rings in division algebras
scientific article; zbMATH DE number 4173161
scientific article