论文标题
在$ ϕ $ - $δ$ - s-s-s-s-primary tomutative戒指中
On $ϕ$-$δ$-S-primary ideals of commutative rings
论文作者
论文摘要
令$ r $为unity $（1 \ not = 0）$的交换戒指，让$ \ mathfrak {j}（r）$是$ r $的所有理想的集合。令$ ϕ：\ mathfrak {j}（r）\ rightarrow \ mathfrak {j}（r）\ cup \ {\ emptySet \} $是$ r $的理想的减少函数，让$Δ：\ mathfrak {j}（j}（j}（j}（r）我们记得，适当的理想$ i $ $ r $称为$ ϕ $ - $δ$ - $ r $的$ r $的理想，如果$ a，b $ in r $ in r $ in r $ in r $ in r $ in i-ϕ（i）$，则是$ a \ in i $或$ a \ in i $或$ b \inΔ（i）$。在本文中，我们介绍了一种新的理想类别，这是对$ ϕ $ - $δ$ - 主要理想的类别的概括。令$ s $为$ r $的多重子集，使得$ 1 \ in s $，而$ i $是$ r $的适当理想，带有$ s \ cap i = \ emptySet $，然后$ i $被称为$ n $ - $ n $ - $Δ$ - $Δ$ - $ s $ - $ s $ s $ - $ r $ in $ s $ in $ s $ in $ a $ a，n e a $ a，b f \ in a b \ in b \ in b \ in a b \ in $ sa \ in I $或$ sb \inδ（i）$。在本文中，我们提出了一系列不同的示例，属性，这一新的理想类别的特征。
Let $R$ be a commutative ring with unity $(1\not=0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $ϕ:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)\cup\{\emptyset\}$ be a reduction function of ideals of $R$ and let $δ:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)$ be an expansion function of ideals of $R$. We recall that a proper ideal $I$ of $R$ is called a $ϕ$-$δ$-primary ideal of $R$ if whenever $a,b\in R$ and $ab\in I-ϕ(I)$, then $a\in I$ or $b\inδ(I)$. In this paper, we introduce a new class of ideals that is a generalization to the class of $ϕ$-$δ$-primary ideals. Let $S$ be a multiplicative subset of $R$ such that $1\in S$ and let $I$ be a proper ideal of $R$ with $S\cap I=\emptyset$, then $I$ is called a $ϕ$-$δ$-$S$-primary ideal of $R$ associated to $s\in S$ if whenever $a,b\in R$ and $ab\in I-ϕ(I)$, then $sa\in I$ or $sb\inδ(I)$. In this paper, we have presented a range of different examples, properties, characterizations of this new class of ideals.
