A new semistar operation on a commutative ring and its applications

Abstract

In this article, a new semistar operation, called the q-operation, on a commutative ring R is introduced in terms of the ring of finite fractions. It is defined as the map by there exists some finitely generated semiregular ideal J of R such that for any where denotes the set of nonzero R-submodules of The main superiority of this semistar operation is that it can also act on R-modules. We can also get a new hereditary torsion theory τq induced by a (Gabriel) topology is an ideal of R with Based on the existing literature of τq-Noetherian rings by Golan and Bland et al., in terms of the q-operation, we can study them in more detailed and deep module-theoretic point of view, such as τq-analog of the Hilbert basis theorem, Krull’s principal ideal theorem, Cartan-Eilenberg-Bass theorem, and Krull intersection theorem.