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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.14829 |
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| _version_ | 1866918001009229824 |
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| author | Blecher, David P. Goswami, Amartya |
| author_facet | Blecher, David P. Goswami, Amartya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathrm{M}$-ideals: from Banach spaces to rings Blecher, David P. Goswami, Amartya Rings and Algebras Operator Algebras 16D25, 16D70, 16D90, 46L05, 46B28 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. |
| title | $\mathrm{M}$-ideals: from Banach spaces to rings |
| topic | Rings and Algebras Operator Algebras 16D25, 16D70, 16D90, 46L05, 46B28 |
| url | https://arxiv.org/abs/2504.14829 |