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The Krull dimension of the semiring of natural numbers is equal to 2

The Krull dimension of the semiring of natural numbers is equal to 2

4 comments

·July 18, 2025
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OgsyedIE

Forgive me for being rusty with this higher level of algebra, but isn't this just counting the degrees of freedom in Spec(Z) and Spec(N) respectively?

ngriffiths

Interesting, makes me curious about geometric ways of looking at semirings. Krull dimension is an algebraic way of capturing the dimension of corresponding geometric objects, so is there some way of doing that with semirings? Or any more intuitive reason why we'd get dimension 2 here? The papers I found in a quick search are way over my head.

macrolocal

By analogy with stacks, my intuition is that Spec(N) still has a one-dimensional geometry, but with a (-1)-dimensional tier from quotienting out the prime points by a semiring action.

MarkusQ

> such that addition distributes over multiplication:

> (a+b)c=ac+bc and c(a+b)=ca+cb

This is usually referred to as multiplication distributing over addition.