Rings and finite fields whose elements are sums or differences of tripotents and potents
We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either exactly seven or strictly more than nine elements, each matrix over $F$ is presentable as a sum of of a tripotent matrix and a $q$-potent matrix if and only if each element in $F$ is presentable as a sum of a tripotent and a $q$-potent, whenever $q>1$ is an odd integer. In addition, if $Q$ is a power of an odd prime and $F$ is a field of odd characteristic, having cardinality strictly greater than $9$, then, for all $n\geq 1$, the matrix ring $\mathbb{M}_n(F)$ is weakly $(Q-1)$-torsion clean if and only if $F$ is a finite field of cardinality $Q$. A novel contribution to the ring-theoretical theme of this study is the classification of finite fields $\FQ$ of odd order in which every element is the sum of a tripotent and a potent. In this regard, we obtain an expression for the number of consecutive triples $γ-1,γ,γ+1$ of non-square elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square elements whenever $\FQ$ contains more than 9 elements.