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Mathematics > Rings and Algebras
Title: Quotient Modules of Finite Length and Their Relation to Fredholm Elements in Semiprime Rings
(Submitted on 14 Apr 2024 (v1), last revised 25 Apr 2024 (this version, v2))
Abstract: 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.
Submission history
From: Niklas Ludwig [view email][v1] Sun, 14 Apr 2024 12:05:16 GMT (9kb)
[v2] Thu, 25 Apr 2024 22:20:15 GMT (9kb)
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