The failure of the Artin-Rees property for the Jacobson radical in prime Noetherian rings.
A two-sided ideal \(I\) of a ring \(R\) is said to satisfy the (weak) right Artin-Rees property if for each right ideal \(E\) of \(R\) there exists an integer \(n\geq 1\) such that \(E\cap I^n\subseteq EI\). In contrast to the commutative case, this property does not hold for most ideals in a noncommutative Noetherian ring \(R\), but for a long time it was hoped that it might hold for the Jacobson radical \(J\) of \(R\), in which case Jacobson's Conjecture \(\bigcap_{n=1}^\infty J^n=0\) would also be true as a consequence.NEWLINENEWLINE In the paper under review, the authors put an end to this hope by constructing a Noetherian prime PI-ring of Krull dimension 3 which is a finitely generated module over its centre and whose Jacobson radical does not satisfy the Artin-Rees property. A semiprime homomorphic image of this ring provides a counterexample of Krull dimension 2. Interestingly, the Artin-Rees property is shown to hold for the Jacobson radical of any Noetherian prime PI-ring which has Krull dimension at most 2, or is equal to its trace ring.
scientific article; zbMATH DE number 6029095
scientific article