Topologically left Artinian rings (Q1086333)

In this paper are considered linearly topologized rings with identity all whose factor-modules at open left ideals are artinian (TA-rings). In the characterization of TA-rings an important role is played by a minimal cogenerator \({}_ RV\) of the class of \(R\)-torsion left \(R\)-modules. Some results about structure and features of the module \({}_ RV\) are achieved. One of the main results of the paper is a theorem describing topologically semilocal TA-rings with topological Jacobson radical finitely generated (such rings have as a basis of neighbourhoods of 0 the closures of powers of the Jacobson radical). For a ring of such type some other results are proved. In the paper are investigated commutative TA-rings and rings, close in their features to TA-rings.