An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group

Let $\mathcal{A}$ be an $\mathbb{F}$-algebra and let $\mathcal{S}$ be its generating set. The length of $\mathcal{S}$ is the smallest number $k$ such that $\mathcal{A}$ equals the $\mathbb{F}$-linear span of all products of length at most $k$ of elements from $\mathcal{S}$. The length of $\mathcal{A}$, denoted by $l(\mathcal{A})$, is defined to be the maximal length of its generating set. In this paper, it is shown that the $l(\mathcal{A})$ does not exceed the maximum of $\dim \mathcal{A}\mathrm{/}2$ and $m(\mathcal{A})-1$, where $m(\mathcal{A})$ is the largest degree of the minimal polynomial among all elements of the algebra $\mathcal{A}$. For arbitrary odd $n$, it is proven that the length of the group algebra of the dihedral group of order $2n$ equals $n$.