$μ$-elements: An extension of essential elements
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916647934099456 |
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| author | Caviglia, Elena Goswami, Amartya Mesiti, Luca |
| author_facet | Caviglia, Elena Goswami, Amartya Mesiti, Luca |
| contents | We introduce and study $μ$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that $μ$-elements are indeed a genuine extension of essential elements. We study preservation of $μ$-elements under contractions and extensions of quantale homomorphisms. We introduce $μ$-complements and $μ$-closedness and study their properties. We determine $μ$-elements for several distinguished quantales, including ideals of $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $μ$-elements in modular quantales. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $μ$-elements: An extension of essential elements Caviglia, Elena Goswami, Amartya Mesiti, Luca Rings and Algebras 06F07, 13A15, 06B23 We introduce and study $μ$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that $μ$-elements are indeed a genuine extension of essential elements. We study preservation of $μ$-elements under contractions and extensions of quantale homomorphisms. We introduce $μ$-complements and $μ$-closedness and study their properties. We determine $μ$-elements for several distinguished quantales, including ideals of $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $μ$-elements in modular quantales. |
| title | $μ$-elements: An extension of essential elements |
| topic | Rings and Algebras 06F07, 13A15, 06B23 |
| url | https://arxiv.org/abs/2503.06739 |