2 problems

Let $d>1$ and let $R$ be a finite commutative ring. For $k\geq 1$, write $S_k(\mathbb{M}_d(R))=\sum_{A\in\mathbb{M}_d(R)}A^k$. The unique element $e\in R\setminus\\{0\\}$ such that…

Non-commutative power-sum conjecture. The theorem for finite commutative rings remains true for non-commutative $R$: $S_k^d(R)=0$ unless $d=2$,…