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[I'm adding a new (non-)answer since what follows has almost nothing to do with my previous one.] Fix a real number $\alpha \ge 1$ and set $S_\alpha := \mathbb N \cup \mathbb R_{\ge \alpha}$. Of cours …

Edit (following the clarifications of the OP): This is not an answer, it is rather a long comment. Studying this kind of questions is part of the mission of factorization theory: The language of the t …

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative Noetheri …

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But …

The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in \ …

In addition to those already suggested by Federico, a good reference is Chapter 2 in Kato's book Perturbation Theory for Linear Operators, especially if you want to tackle the question from the higher …