binomial theorem | Problems in Mathematics

10/18/2017

Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues

Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$.
Assume that $A-B$ is a nilpotent matrix.

Then prove that the eigenvalues of $A$ and $B$ are the same.

04/02/2017

Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have
\[I_1+I_2=R.\]

Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.