Bi-Artinian Noetherian rings (Q5890196)

A ring \(R\) is right (left) semi-Noetherian if each two-sided ideal of \(R\) is right (left) finitely generated. The author proves the following main results: (i) A semi-Noetherian fully right Goldie ring \(R\) satisfies the descending chain condition (\(R\) is bi-Artinian) on two-sided ideals if and only if for every prime ideal \(P\), \(R/P\) has a unique minimal ideal; (ii) Let \(R\) be a semi-Noetherian fully right Goldie ring such that \(R\) is not bi-Artinian but for any essential ideal \(E\), \(R/E\) is bi-Artinian. Let also the Jacobson radical \(J(R)\subseteq A\), where \(A=A(R)\) be the intersection of all quasi-primitive ideals of \(R\). Then \(\bigcap_{n\in\mathbb{N}}J^n=\bigcap_{n\in\mathbb{N}}A^n=0\) if either \(R\) is semiprime, or \(R\) is fully left Goldie.