Left and right associated prime ideals in chain rings with d.c.c. for prime ideals (Q1099236)

A ring R (with identity) is a chain ring if its lattice of right ideals as well as its lattice of left ideals are totally ordered by inclusion. Let \(I\neq R\) be a two-sided ideal in a chain ring R and \(S_{\ell}(I)=\{s\in R|\) st\(\in I\) implies \(t\in I\}\) with \(S_ r(I)\) defined similarly. It follows that \(P_{\ell}(I)=R\setminus S_{\ell}(I)\) is a completely prime ideal. In general \(P_{\ell}(I)\neq P_ r(I)\), but it is proved here that \(P_{\ell}(I)=P_ r(I)\) provided that the minimum condition holds for prime ideals that contain I. Conditions on the prime ideals associated with an ideal I in a chain ring R play a role in Hjelmslev geometries. (There is a star missing in the definition of \(P_{\ell}(I)\) in the introduction).