Hereditary orders with infinitely many idempotent ideals
In the classical theory of orders over a Dedekind domain, any hereditary order R is equivalent to a maximal order S, and R can be obtained from S as a finitely iterated idealizer. That a finite sequence of iterations suffices to get from S to R depends on R having only finitely many idempotent ideals. In the general theory of orders in simple artinian rings (in the sense of Goldie), the analogs of hereditary orders are hereditary noetherian prime (HNP) rings, and the analogs of maximal orders are Dedekind prime rings. \textit{J. C. Robson} proved that an HNP ring R is equivalent to a Dedekind prime ring if and only if R has only finitely many idempotent ideals, in which case R can be obtained from the Dedekind prime ring as a finitely iterated idealizer [J. Algebra 22, 45- 81 (1972; Zbl 0239.16003), Theorem 6.3]. Whether or not every HNP ring has this form remained an open question, which the present authors settle in the negative by constructing HNP rings with infinitely many idempotent ideals. Variations in their construction produce examples in which (a) all the idempotent maximal ideals belong to cycles of length two, or (b) all the idempotent maximal ideals belong to cycles but there is one cycle of each prime integer length, or (c) none of the idempotent maximal ideals belong to cycles. The present expository paper provides an overview of the construction process, which is a special type of infinitely iterated idealizer. Details will appear in a longer paper of the authors [''Constructions of hereditary noetherian rings and simple rings'', to appear in Proc. Lond. Math. Soc., III Ser.].
scientific article; zbMATH DE number 3842067
scientific article