AI 中文总结
该研究刻画了含单位元交换环上多项式环的本质理想，通过构造测试类多项式判定本质性，针对不同类环细化测试类并研究了阿廷环等特殊情形。
AI 中文摘要
我们对含单位元的交换环$R$上的多项式环$R[(X_λ)_{λ∈Λ}]$中的本质理想进行刻画，针对$R$的多个知名类环给出了若干种刻画方式。核心思路是通过检验与测试类多项式生成的主理想的交集来判定本质性。在一般交换环情形下，我们运用McCoy零因子判据，基于内容理想的零化子构造了高效的测试类多项式；随后，在系数环$R$满足更多性质的假设下，我们对测试类进行了细化并强化了结果：当$R$为诺特环时，借助$R$的相伴素元简化测试类；当$R$满足Serre的$(S_1)$条件时，仅需使用极小素元即可；此外，我们还研究了$R$为阿廷环、$R=\boldsymbol{Z}/n\boldsymbol{Z}$以及$R$为优秀环的理想-adic完备化等情形。
英文摘要
We characterize essential ideals in polynomial rings $R[(X_λ)_{λ\inΛ}]$ over a commutative ring $R$ with $1$. We present several characterizations, letting $R$ vary among notable classes of rings. The idea is to detect essentiality by checking the intersections with principal ideals generated by polynomials in a test class. In the general commutative case, we apply McCoy's zero-divisor criterion to construct an efficient test class of polynomials in terms of annihilators of content ideals. We then refine the test class and strengthen the result in several ways, assuming further properties on the coefficients ring $R$. Assuming that $R$ is Noetherian, we reduce the test class using the associated primes of $R$. When $R$ satisfies Serre's condition $(S_1)$, it suffices to use minimal primes. We also study the cases of $R$ Artinian, $R=\mathbb{Z}/n\mathbb{Z}$ and $R$ equal to the ideal-adic completion of an excellent ring, among others.
