On a Generalized Finite Intersection Property

Abstract

We continue the study of the right finite intersection property under a weaker condition on annihilators, introducing the concept of generalized right finite intersection property (simply, generalized right FIP). We observe the structure of rings with the generalized right FIP and examine the generalized right FIP for various kinds of basic extensions of rings with the property. We show that the generalized right FIP does not go up to polynomial rings, and that the 2-by-2 full matrix ring over a domain has the generalized right FIP. In the process, we also obtain an equivalent condition for which a nonzero polynomial, over the ring of integers modulo n ≥ 2, is a non-zero-divisor.

Key Words:

• Domain
• Generalized right FIP
• Matrix ring
• Polynomial ring
• Right FIP

2010 Mathematics Subject Classification:

• 16P99
• 16S36
• 16S50

ACKNOWLEDGMENTS

The authors thank the referee for his/her very careful reading of the manuscript and valuable suggestions that improved the article. The third named author was supported for a sabbatical year by Gyeongsang National University Research Grant. The fourth named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20110004745). The third-named author also really thanks Professor John Clark and Mrs. Austina for all the hospitality during his stay in University of Otago.

Notes

Communicated by J. Zhong.