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 $Q_0(R)$ of finite fractions. It is defined as the map $q:{\mathcal{F}}_q(R)\to {\mathcal{F}}_q(R)$ by $A\mapsto A_q:=\{x\in Q_0(R)|$ there exists some finitely generated semiregular ideal J of R such that $Jx\subseteq A\}$ for any $A\in {\mathcal{F}}_q(R),$ where ${\mathcal{F}}_q(R)$ denotes the set of nonzero R-submodules of $Q_0(R).$ 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 τ_q induced by a (Gabriel) topology $\left\{I\right|I$ is an ideal of R with $I_q=R_q\}.$ Based on the existing literature of τ_q-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 τ_q-analog of the Hilbert basis theorem, Krull’s principal ideal theorem, Cartan-Eilenberg-Bass theorem, and Krull intersection theorem.

Keywords:

• Cartan-Eilenberg-Bass theorem
• Hilbert basis theorem
• Krull intersection theorem
• Krull’s principal ideal theorem
• Lucas module
• τ_q-Noetherian ring
• q-operation
• semiregular ideal
• the ring of finite fractions

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C99
• 13A15

Acknowledgements

The authors sincerely thank the referee for several useful comments.

Additional information

Funding

The study was funded by the National Natural Science Foundation of China (No. 11671283) and the doctoral foundation of Southwest University of Science and Technology (No. 17zx7144).