On $S$-Noetherian Lattices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917444995514368 |
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| author | Sarode, Sachin Patil, Chetan Joshi, Vinayak |
| author_facet | Sarode, Sachin Patil, Chetan Joshi, Vinayak |
| contents | In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish a Cohen-Kaplansky type theorem for $S$-Noetherian lattices, showing that $L$ is $S$-Noetherian if and only if every $S$-prime element of $L$ is $S$-compact. Finally, we introduce the concept of $S$-primary elements-a generalization of primary elements in multiplicative lattices and demonstrate the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26058 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $S$-Noetherian Lattices Sarode, Sachin Patil, Chetan Joshi, Vinayak Commutative Algebra Primary 06F10, 13A15, 13C05, Secondary 06A11 In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish a Cohen-Kaplansky type theorem for $S$-Noetherian lattices, showing that $L$ is $S$-Noetherian if and only if every $S$-prime element of $L$ is $S$-compact. Finally, we introduce the concept of $S$-primary elements-a generalization of primary elements in multiplicative lattices and demonstrate the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian lattices. |
| title | On $S$-Noetherian Lattices |
| topic | Commutative Algebra Primary 06F10, 13A15, 13C05, Secondary 06A11 |
| url | https://arxiv.org/abs/2604.26058 |