标题： $\mathrm{M}$-理想：从巴拿赫空间到环 显示英文标题

标题： $\mathrm{M}$-ideals: from Banach spaces to rings

摘要： 我们引入并研究了一类环理想，称为环 $\mathrm{M}$-理想，这些概念受到Alfsen–Effros在Banach空间中关于 $\mathrm{M}$-理想的理论的启发。我们证明了 $\mathrm{M}$-理想推广了经典的本质理想的概念，并将其作为子类包含其中。中心定理给出了完整的刻画：一个理想是 $\mathrm{M}$-理想当且仅当它要么是本质的理想，要么是相对不可约的。这一二分法揭示了 $\mathrm{M}$-理想的丰富性和多样性，包括本质理想和极小理想，并在非交换和非幺元环中自然推广。我们系统地研究了 $\mathrm{M}$-理想在诸如交、商、直积以及Morita等价的标准构造下的代数稳定性，并在拓扑环和算子代数中确立了它们的行为。 在某些环（如$\mathbb{Z}_n$）和C*-代数中，我们完全分类了$\mathrm{M}$-理想，并将其与代数最小投影和中心幂等元联系起来。 证明了$\mathrm{M}$-理想在$C(K)$中恰好是本质理想或对应于孤立点的极小理想。 从结构上看，我们证明了没有适当$\mathrm{M}$-理想的缺失表征了简单性，而每个适当$\mathrm{M}$-理想都是直和项的环必须分解为有限个简单环的直和。 最后，我们引入了$\mathrm{M}$-补的概念，类比模论中的本质扩张，并证明了它们的存在性。 显示英文摘要

摘要： We introduce and investigate a class of ring ideals, termed ring $\mathrm{M}$-ideals, inspired by the Alfsen--Effros theory of $\mathrm{M}$-ideals in Banach spaces. We show that $\mathrm{M}$-ideals extend the classical notion of essential ideals and subsume them as a subclass. The central theorem provides a full characterization: an ideal is an $\mathrm{M}$-ideal if and only if it is either essential or relatively irreducible. This dichotomy reveals the abundant and diverse nature of $\mathrm{M}$-ideals, encompassing both essential and minimal ideals, and admits natural generalizations in rings beyond the commutative and unital settings. We systematically study the algebraic stability of $\mathrm{M}$-ideals under standard constructions such as intersection, quotient, direct product, and Morita equivalence and establish their behavior in topological rings and operator algebras. In certain rings such as $\mathbb{Z}_n$ and C*-algebras, we completely classify $\mathrm{M}$-ideals and relate them to algebraically minimal projections and central idempotents. The ring $\mathrm{M}$-ideals in $C(K)$ are shown to be precisely the essential ideals or those minimal ideals corresponding to isolated points. Structurally, we show that the absence of proper $\mathrm{M}$-ideals characterizes simplicity, while rings in which every proper $\mathrm{M}$-ideal is a direct summand must decompose as finite direct sums of simple rings. In closing, we introduce the notion of $\mathrm{M}$-complements, drawing an analogy with essential extensions in module theory, and demonstrate their existence.