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.
Acknowledgements
The authors sincerely thank the referee for several useful comments.