On the Polya Permanent Problem over Finite Commutative Rings

In this paper we address the Polya permanent problem that was first raised in the second decade of the last century. Despite this, it continues to be treated in several surveys, of which we highlight the studies that point out Polya’s permanent problem over finite fields. Unlike previous papers, we focus on finite commutative rings, and to this end, we start by considering a commutative ring with identity <I>R</I> and its decomposition into a direct sum of finite local rings. Next we suppose that the characteristic of each residue field Fiis different from two, and we proof that if <i>n</i> is greater than or equal to 3, then no bijective map Φ from <i>M<sub>n</sub></i>(<I>R</I>) to <i>M<sub>n</sub></i>(<I>R</I>) transforms the permanent into a determinant. We developed this technique to estimate the order of the general linear group of degree <i>n</i> over a finite commutative ring with identity. The paper begins with the introduction where we present the title, the preliminaries that help the understanding of the following subject, then we talk about the unit permanent and unit determinant in <i>M<sub>n</sub></i>(<I>R</I>), we demonstrate the main result and conclusions. Regarding the methodology, we use the previous results on finite fields and the structure of finite commutative rings and also radical theory of rings.