Finite superideal domains (Q2822565)

As defined by the authors, an integral domain \(R\) is a {finite superideal domain (FSD)} if each nonzero ideal of \(R\) is contained in just finitely many ideals of \(R\). An FSD is Noetherian and of dimension \(\leq1\). Dedekind and Cohen-Kaplansky domains are FSD's, but there are FSD's there are neither Dedekind, nor Cohen-Kaplansky. An FSD is an FFD (finite factorization domain) and for one-dimensional local domains these two properties are equivalent. The authors provide several characterizatons of FSD's. For example, an integral domain \(R\) is an FSD if and only if for each nonzero ideal \(I\) of \(R\), the ring \(R/I\) is a finite direct product of SPIR's and a finite ring (recall that a SPIR (a special principal ideal ring) is a local principal ideal ring with nilpotent maximal ideal). The class of FSD's is closed under localizations. The authors characterize when a ring \(R=D+M\) is an FSD, where \(D\) is a proper subring of a field \(K\) such that \(K+M\) is a valuation domain.with maximal ideal \(M\). By this characterization, \(R=D+M\) is an FSD if and only if \(R\) is an FFD. The authors also deal with the FSD property of monoid domains, etc.