4 problems

Let $R$ be a discrete valuation ring (DVR) with fraction field $K$, and let $A$ and $B$ be $n\times n$ matrices over $R$. Suppose that $L/K$ is a finite extension and $S$ is the in…

Let $R$ be a discrete valuation ring with residue field $\mathbf F_q$, let $P$ be its maximal ideal, and let $M_m(R/P^n)$ denote the ring of $m\times m$ matrices over $R/P^n$. An…

Semisimple similarity number conjecture. For arbitrary $m\times m$ sequences, not necessarily semisimple, $\mathcal{A}\sim\mathcal{B}$ if and only if…

Let $A$ be the local principal ideal ring under consideration, let $\alpha\in\operatorname{M}_n(A)$, and for each $m=1,\ldots,n$ let the $m$-th Fitting ideal of $\alpha$ be the ide…