Rings and finite fields whose elements are sums or differences of tripotents and potents (2112.14617v3)

Abstract: 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 $\gamma-1,\gamma,\gamma+1$ of non-square elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square elements whenever $\FQ$ contains more than 9 elements.