Quotient Modules of Finite Length and Their Relation to Fredholm Elements in Semiprime Rings

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 that the A-modules of the form A/A a and A/aA are of finite length. We are particularly interested in sufficient conditions for our generalized Fredholm elements to be Fredholm. In a unital C^*-algebra A we shall even see that an element a is Fredholm if and only if the A-modules A/A a and A/aA both have finite length.