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The Krull dimension of the semiring of natural numbers is equal to 2
Let $R$ be a ring. Its Krull dimension is the supremum of the lengths $n$ of chains $P_0\subsetneq P_1 \subsetneq\dots\subsetneq P_n$ of pri...
The amusing part comes from the classification of prime and maximal ideals of the semiring $\mathbf N$ of natural numbers, which I learned of via a Lean formalization project led by Junyan Xu. One direction is clear, if $n$ and $ab-a-b-n$ can both be written in this form, then so can their sum, which is $ab-a-b$, a contradiction. Another open question of the style of the proposition had been raised by Frobenius: consider mutually coprime integers $a_1,\dots,a_r$; then any large enough integer $n$ can be written as $n=a_1u_1+\dots+a_ru_r$, for some natural numbers $u_1,\dots,u_r$, but when $r\geq 3$, there is no known formula for largest natural number that cannot be written in this form.
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