Torsion functors with monomial support (Q361251)

Let \(R\) be a commutative ring with identity and \(\mathfrak a,\) \(\mathfrak b\) be two ideals of \(R\). Recall that the \(\mathfrak a\)-torsion functor \(\Gamma_{\mathfrak a}(-)\) is a subfunctor of the identity functor which is defined by \(\Gamma_{\mathfrak a}(M):=\bigcup_{n\in\mathbb{N}}(0:_M{\mathfrak a}^n)\) for all \(R\)-modules \(M\). When \(R\) is Noetherian, it is an easy exercise to verify that the two functors \(\Gamma_{\mathfrak a}(-)\) and \(\Gamma_{\mathfrak b}(-)\) are the same if and only if \(\sqrt{\mathfrak a}= \sqrt{\mathfrak b}\). In the non-Noetherian case the situation is completely different. Let \(\mathfrak a\) and \(\mathfrak b\) be two monomial ideals of a not necessarily Noetherian polynomial algebra \(T\). In the paper under review, the author investigates the question when \(\Gamma_{\mathfrak a}(-)=\Gamma_{\mathfrak b}(-)\). One of his main results asserts that if \(\mathfrak a\) and \(\mathfrak b\) are of finite type, then \(\Gamma_{\mathfrak a}(-)= \Gamma_{\mathfrak b}(-)\) if and only if \(\sqrt{\mathfrak a}=\sqrt{\mathfrak b}\).