On weakly 2-prime ideals in commutative rings

Abstract

The purpose of the paper is to introduce and study weakly 2-prime ideals in commutative rings. Let A be a commutative ring with a nonzero identity. A proper ideal P of A is said to be a weakly 2-prime ideal if whenever $0\ne xy\in P$ for some $x,y\in A,$ then $x^2\in P$ or $y^2\in P.$ Besides giving various examples and characterizations of weakly 2-prime ideals, we investigate the relations between weakly 2-prime ideals and other classical ideals such as 2-prime ideals, weakly prime ideals and weakly 2-absorbing primary ideals. Furthermore, we study 2-prime avoidance lemma in commutative rings and introduced the class of compactly 2-packed rings. Finally, we investigate the compactly packedness, compactly 2-packedness and coprimely packedness of trivial extension $A⋉M$ of an A-module M.

Keywords:

• Compactly 2-packed ring
• compactly packed ring
• coprimely packed ring
• 2-prime ideal
• weakly 2-prime ideal
• weakly prime ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13C05
• 13G05

Acknowledgments

The author would like to thank the referee for his/her great efforts in proofreading the manuscript.