AI 中文总结
该研究提出有限交换环的理想零乘积概率不变量，推导其在各类环上的显式公式，分析其性质与渐近行为，并与环元素的经典零乘积概率作对比，揭示二者结构信息的差异。
AI 中文摘要
我们引入理想零乘积概率（ideal zero-product probability），这是与有限交换环$R$相关的一种新的概率不变量，定义为：$\zeta_k(R) = \frac{ |\{(I_1,\ldots,I_k)\in\mathcal I(R)^k: I_1\cdots I_k=(0)\}| } {|\mathcal I(R)|^k}$，其中$\mathcal I(R)$表示$R$的所有理想构成的集合。该量给出了$k$个独立且均匀选取的理想的乘积为零理想的概率。我们确定了其基本性质，包括在环同构下的不变性、随$k$的单调性以及关于有限直积的可乘性。针对有限域、有限布尔环、有限链环和有限主理想环，我们得到了显式公式。对于有限链环，我们证明该不变量仅依赖于Loewy长度。我们还利用容斥原理、有界分拆以及生成函数推导了闭式公式。最后，我们研究了$\zeta_k(R)$的渐近行为，证明当$k\to\infty$时它收敛到1。我们还将理想零乘积概率与有限链环上环元素的经典零乘积概率进行了比较，得到了显式公式，并表明这两种不变量捕捉了根本不同的结构信息。
英文摘要
We introduce the \emph{ideal zero-product probability}, a new probabilistic invariant associated with a finite commutative ring $R$, defined by \[ ζ_k(R) = \frac{ |\{(I_1,\ldots,I_k)\in\mathcal I(R)^k : I_1\cdots I_k=(0)\}| } {|\mathcal I(R)|^k}, \] where $\mathcal I(R)$ denotes the set of all ideals of $R$. This quantity gives the probability that the product of $k$ independently and uniformly chosen ideals is the zero ideal. We establish its basic properties, including invariance under ring isomorphisms, monotonicity in $k$, and multiplicativity with respect to finite direct products. Explicit formulas are obtained for finite fields, finite Boolean rings, finite chain rings, and finite principal ideal rings. For finite chain rings, we prove that the invariant depends only on the Loewy length. We also derive closed formulas using inclusion--exclusion and bounded compositions, together with generating functions. Finally, we study the asymptotic behaviour of $ζ_k(R)$ and prove that it converges to $1$ as $k\to\infty$. We also compare the ideal zero-product probability with the classical zero-product probability of ring elements on finite chain rings, obtaining explicit formulas and showing that the two invariants capture fundamentally different structural information.
Comments33 pages, comments are welcome
