Quotient Modules of Finite Length and Their Relation to Fredholm Elements in Semiprime Rings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910423917264896 |
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| author | Ludwig, Niklas |
| author_facet | Ludwig, Niklas |
| contents | B. A. Barnes introduced so-called Fredholm elements in a semiprime ring whose definition is inspired by Atkinson's theorem. Here the socle of a semiprime ring generalizes the ideal of finite-rank operators on a Banach space. In this paper, we aim to see that the algebraic concept of the length of a module is strongly related to that of Fredholm elements. This motivates another generalization of Fredholm elements by requiring for an element $a\in\mathcal{A}$ that the $\mathcal{A}$-modules of the form $\mathcal{A}/\mathcal{A} a$ and $\mathcal{A}/a\mathcal{A}$ are of finite length. We are particularly interested in sufficient conditions for our generalized Fredholm elements to be Fredholm. In a unital C$^*$-algebra $\mathcal{A}$ we shall even see that an element $a\in\mathcal{A}$ is Fredholm if and only if the $\mathcal{A}$-modules $\mathcal{A}/\mathcal{A} a$ and $\mathcal{A}/a\mathcal{A}$ both have finite length. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quotient Modules of Finite Length and Their Relation to Fredholm Elements in Semiprime Rings Ludwig, Niklas Rings and Algebras Functional Analysis Operator Algebras 16P70, 46H05, 46H10, 46L05, 47A53 B. A. Barnes introduced so-called Fredholm elements in a semiprime ring whose definition is inspired by Atkinson's theorem. Here the socle of a semiprime ring generalizes the ideal of finite-rank operators on a Banach space. In this paper, we aim to see that the algebraic concept of the length of a module is strongly related to that of Fredholm elements. This motivates another generalization of Fredholm elements by requiring for an element $a\in\mathcal{A}$ that the $\mathcal{A}$-modules of the form $\mathcal{A}/\mathcal{A} a$ and $\mathcal{A}/a\mathcal{A}$ are of finite length. We are particularly interested in sufficient conditions for our generalized Fredholm elements to be Fredholm. In a unital C$^*$-algebra $\mathcal{A}$ we shall even see that an element $a\in\mathcal{A}$ is Fredholm if and only if the $\mathcal{A}$-modules $\mathcal{A}/\mathcal{A} a$ and $\mathcal{A}/a\mathcal{A}$ both have finite length. |
| title | Quotient Modules of Finite Length and Their Relation to Fredholm Elements in Semiprime Rings |
| topic | Rings and Algebras Functional Analysis Operator Algebras 16P70, 46H05, 46H10, 46L05, 47A53 |
| url | https://arxiv.org/abs/2404.09224 |