On DICC rings (Q1084457)

An R-module M, over a commutative ring R, is said to satisfy the doubly infinite chain condition (DICC) if any infinite chain of submodules of M stabilizes either to the right or to the left (or to both sides). A ring which satisfies the DICC condition as a module is called a DICC ring. The main result of this paper is that a DICC ring is either Noetherian or a direct product of an Artinian ring, and a non-Noetherian ring S with no prime ideal both minimal and maximal, and with the following four properties: (1) \(S_{red}\) is Noetherian; (2) N (the nilradical of S) is nilpotent; (3) N has DCC; (4) for each \(x\in S-N\), \(N/Sx\cap N\) has finite length.