On rings and modules with DICC
A module over a commutative ring R has the property DICC if every chain of submodules indexed by the integers terminates to one side or the other. The author characterizes such modules as those modules M having an Artinian submodule N such that M/N is Noetherian and N/Rx\(\cap N\) is of finite length for all \(x\in M\setminus N\). A construction of rings that have DICC as modules over themselves is given. It is shown that if M has DICC, but is neither Noetherian nor Artinian, and if no maximal ideal is minimal in the support of M, then the support of M is closed and irreducible in Spec(R) Some of the proofs use an interesting concept called almost-divisibility. The R-module Q is almost-divisible if Q/aQ has finite length, for all a not in the nilradical of R.
scientific article; zbMATH DE number 3946298