论文标题
在弱$（m，n）上 - $关闭$δ-$的通勤戒指的主要理想
On weakly $(m,n)-$closed $δ-$primary ideals of commutative rings
论文作者
论文摘要
令$ r $为$ 1 \ neq0 $的交换戒指。在本文中，我们介绍了$ $（m，n）的概念 - $ $ close $δ-$ $ r $的主要理想，并探索其基本属性。我们表明，$ i \ bowtie^{f} j $是一个弱$（m，n） - $关闭$δ_ {\ bowtie^{f}} - $ $ a $ a \ a \ bowtie^{f} j $的主要理想不是$（m，n） - $ nive $Δ__$ $ $(m,n)-$closed $δ-$primary ideal of $A$ that is not $(m,n)-$closed $δ-$primary and for every $δ$-$(m,n)$-unbreakable-zero element $a$ of $I$ we have $(f(a)+j)^{m}=0$ for every $j\in~J$, where $f:A\rightarrow B$ is a homomorphism戒指和$ j $是$ B的理想。$此外，我们提供了示例以证明结果的有效性和适用性。
Let $R$ be a commutative ring with $1\neq0$. In this article, we introduce the concept of weakly $(m,n)-$closed $δ-$primary ideals of $R$ and explore its basic properties. We show that $I\bowtie^{f}J$ is a weakly $(m,n)-$closed $δ_{\bowtie^{f}}-$primary ideal of $A\bowtie^{f}J$ that is not $(m,n)-$closed $δ_{\bowtie^{f}}-$primary if and only if $I$ is a weakly $(m,n)-$closed $δ-$primary ideal of $A$ that is not $(m,n)-$closed $δ-$primary and for every $δ$-$(m,n)$-unbreakable-zero element $a$ of $I$ we have $(f(a)+j)^{m}=0$ for every $j\in~J$, where $f:A\rightarrow B$ is a homomorphism of rings and $J$ is an ideal of $B.$ Furthermore, we provide examples to demonstrate the validity and applicability of our results.
