关于Anderson-Badawi $ω_{R[X]}(I[X])=ω_R(I)$猜想
On the Anderson-Badawi $ω_{R[X]}(I[X])=ω_R(I)$ conjecture
摘要：
设$R$是单位元不为零的交换环，$n$是正整数. Anderson和Badawi在他们关于$n$ -吸收理想的论文中，定义了交换环$R$的真理想$I$是$R$的$n$ -吸收理想，如果无论何时对于$x_1, \ldots, x_{n+1} \in R$有$x_1 \cdots x_{n+1} \in I$，则存在$x_i$的$n$，其乘积在$I$中，并猜想对于任意环$R$的任何理想$I$有$ω_{R[X]}(I[X])=ω_R(I)$，其中$ω_R(I)= \min \{n\colon\text{$ I $ is an $ n $-absorbing ideal of $ R $}\}$。在本文中，我们使用内容公式技术证明了他们的猜想是正确的，如果下列条件之一成立：环$R$是Prüfer整环.环$R$是高斯环，使得它的加群是无挠的。环$R$的加群是无挠的，$I$是$R$的根理想。
英文摘要：
Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \cdots x_{n+1} \in I$ for $x_1, \ldots, x_{n+1} \in R$, then there are $n$ of the $x_i$'s whose product is in $I$ and conjecture that $ω_{R[X]}(I[X])=ω_R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $ω_R(I)= \min \{n\colon\text{$I$ is an $n$-absorbing ideal of $R$}\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Prüfer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
