Further characterizations of Artinian rings. (Q1882126)

The author proves that the following four conditions on the ring \(R\) with prime radical \(N\) are equivalent. a) \(R\) is right Artinian. b) \(R\) has Krull dimension as right \(R\)-module and there does not exist a finitely generated right \(R\)-module \(M\) with \(\text{ann}(M)\) a prime ideal that is torsion with respect to the set of regular elements of \(R\) modulo \(N\). c) \(R\) is semiprimary with \(R\) as right \(R\)-module of finite reduced rank. d) Every right \(R\)-module \(M\) has a finite subset \(X\) such that \(\text{ann}(X)=\text{ann}(M)\).