论文标题
在弱的三角中，可交换环的理想
On weakly delta-semiprimary ideals of commutative rings
论文作者
论文摘要
令$ r $为$ 1 \ neq 0 $的交换戒指。我们记得，如果$ a，b $ in r $ in r $，则称为$ r $的适当理想$ i $ $ r $，in i $，然后是$ a \ in \ sqrt {i} $或$ b \ in \ sqrt {i} $。我们说$ i $是$ r $的{\ in {\ in r $ in r $ in r $ n时，$ 0 \ not = ab \ in i $，然后$ a \ in \ sqrt {i} $或$ b \ in \ sqrt {i} $。在本文中，我们介绍了一种新的理想类别，这些理想与（弱）半二级理想阶级密切相关。令$ i（r）$为$ r $的所有理想的集合，让$δ：i（r）\ rightarrow i（r）$成为一个函数。然后，$δ$称为$ r $的理想的扩展功能，如果$ l，i，j $是$ r $的理想，那么$ j \ subseteq i $，然后是$ l \subseteqΔ（l）$和$Δ（j）$sebseteqΔ（j）fsebSeteqΔ（i）$。令$δ$为$ r $的理想的扩展功能。然后，一个适当的理想$ i $ $ $ r $（即$ i \ not = r $）称为a（{{\ it $δ$ -semiprimary}）{\ ut弱$ $Δ$ -semiprimary} $ r $ $ $ $ $ If（$ ab b \ in $ ab in $ $ 0 $ $ ab $ ab ab $ $ ab b y y $ ab b y in $ a $ a a a a a i $ a a） δ（i）$。例如，令$δ：i（r）\ rightarrow i（r）$，这样$δ（i）= \ sqrt {i} $。然后，$δ$是$ r $的理想的扩展功能，因此，适当的理想$ i $ $ r $是（$δ$ - iPimiprimary）弱$δ$Δ$ -smiprimary $ r $的理想，并且只有$ i $是a（半利润）弱的弱点$ r $。给出了许多有关$δ$ - 占主导地位的理想和弱$δ$ - 含量理想示例的结果。
Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly semiprimary ideal} of $R$ if whenever $a,b\in R$ and $0 \not = ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let $I(R)$ be the set of all ideals of $R$ and let $δ: I(R) \rightarrow I(R)$ be a function. Then $δ$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of $R$ with $J \subseteq I$, then $L \subseteq δ(L)$ and $δ(J) \subseteq δ(I)$. Let $δ$ be an expansion function of ideals of $R$. Then a proper ideal $I$ of $R$ (i.e., $I \not = R$) is called a ({\it $δ$-semiprimary}) {\it weakly $δ$-semiprimary} ideal of $R$ if ($ab \in I$) $0 \not = ab \in I$ implies $a \in δ(I)$ or $b \in δ(I)$. For example, let $δ: I(R) \rightarrow I(R)$ such that $δ(I) = \sqrt{I}$. Then $δ$ is an expansion function of ideals of $R$ and hence a proper ideal $I$ of $R$ is a ($δ$-semiprimary) weakly $δ$-semiprimary ideal of $R$ if and only if $I$ is a (semiprimary) weakly semiprimary ideal of $R$. A number of results concerning weakly $δ$-semiprimary ideals and examples of weakly $δ$-semiprimary ideals are given.
