Topologies and rings which arise from Artinian modules over a commutative ring (Q1903033)

Let \(R\) be a commutative ring with identity, \(M\) be a certain unital \(R\)- module and \(I\) an ideal in \(R\). -- A linear topology \({\mathcal D}\) on \(R\) is called: (1) Artinian, if \(R/J\) is Artinian for every open ideal \(J\) in \(R\); (2) \(M\)-topology, if the set \(\{(0 :_R N) |N\) is a finitely generated submodule of \(M\}\) forms a base of neighbourhoods of zero in \({\mathcal D}\); (3) weakly adic, if there exists an ideal \(J\) in \(R\) such as the set of all subsets, every of which is a closure of some set \(J^n\), forms a base of neighbourhoods of zero in \({\mathcal D}\). It is proved that there exists an \(R\)-module \(X\) such that: -- a topology \({\mathcal D}\) is an Artinian topology iff it is an \(M\)- topology for some submodule \(M\) of \(X\); -- an Artinian topology \({\mathcal D}\) is weakly adic iff the corresponding module \(M\) is Artinian. Some properties of a completion of a topological ring equipped with weakly adic or \(M\)-topology are investigated.