# Nil ring version of Burnside's problem

The famous Burnside group of exponent $n$ over a set $G$ of generators is the group defined by generators $g\in G$ and relations $h^n=1$ where $h$ is any product of elements of $G$.

Likewise, let $G$ be any set of indeterminates, let ${\mathbb Q}[G]$ denote the ring of noncommutative polynomials in $G$, and let ${\mathbb Q}'[G]$ denote the hyperplane in ${\mathbb Q}[G]$ of polynomials with no constant term. Denote by $I_n$ the ideal generated by all $n$-th powers of elements of ${\mathbb Q}'[G]$. Then $B_n(G)={\mathbb Q}'[G]/I$ might be called the "Burnside ring" of exponent $n$ over $G$.

When $n=2$, $B_2(G)$ is simply the exterior algebra over $G$.

Is it true that $B_n(G)$ is always finite dimensional when $G$ is finite ? Is a (more or less) simply described basis known ?

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