A note on an inverse problem in algebra

TL;DR

This paper examines the ill-posed nature of reconstructing semisimple group algebras from specific matrix rings over finite fields and proposes a new concept of completeness to address this issue.

Contribution

It introduces the concept of completeness for certain rings to transform an ill-posed inverse problem into a well-posed one and proposes a related conjecture.

Findings

Identifies the inverse problem as ill-posed

Defines the concept of completeness for rings

Proposes a conjecture to achieve well-posedness

Abstract

In this paper, we discuss the inverse problem of determining a semisimple group algebra from the knowledge of rings of the type sum_{t=1}^s M_{n_t}(Ft), where j is an arbitrary integer and F_t is finite field for each t, and show that it is ill-posed. After then, we define the concept of completeness of the rings of the type sum_{t=1}^s M_{n_t}(Ft) to pose a well-posed inverse problem and propose a conjecture in this direction.