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.

摘要

本文的目的是介绍和研究交换环中的弱二素理想。设A是一个具有非零身份的交换环。一个适当的理想P的一个被认为是弱2贷理想的，如果只要$0\ne X是\in 磷$ 对于一些 $X,是\in \mathrm{一种},$ 然后 $X^2\in 磷$ 或者 ${是}^2\in 磷.$除了给出弱二素理想的各种例子和刻画外，我们还研究了弱二素理想与其他经典理想如二素理想、弱素理想和弱吸收二元理想之间的关系。此外，我们研究了可交换环中的 2-prime 回避引理，并介绍了紧凑 2-packed 环的类。最后，我们研究了平凡扩展的紧包装性、紧致 2-包装性和互质性包装性$\mathrm{一种}⋉米$一个A模块M。