There are already a lot of discussion about the motivation for prime spectrum of commutative ring. In my perspective(highly non original), there are following reasons for the importance of prime spectrum.
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$\text{Spec}(R)$ is the minimal spectrum containing $\text{Spec}_{\rm max}(R)$ which has good functoriality which means the preimage of a prime ideal is still a prime ideal.
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if $p\in \text{Spec}(R)$, then $S_{p} = R-p$ is multiplicative set. Then one can localize.
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$S_{p}^{-1}R$ for $p\in \text{Spec}(R)$ is a local ring (has unique maximal ideal which is equivalent to have unique isomorphism class of simple modules). Local ring is easy to deal with and the maximal ideal can be described in explicitly, i.e $m=S_p^{-1}p$
(My advisor told me P.Cartier pushed Grothendieck to built commutative algebraic geometry machinery based on prime spectrum and the reasons mentioned above are the reasons they used prime spectrum)
Addtional reason: one can have good definitions of topological space and a structure sheaf on it so that one can recover this commutative ring back as its global section
Now, my question is for the people coming from commutative world, what other properties do you expect the spectrum of a noncommutative ring should have?
I am aware that people are coming from different branches, there might be various kinds of noncommutative ring arising in your study. Therefore, the question for people coming from different branches of mathematics is that which kind of noncommutative ring do you meet and what properties do you feel that the spectrum of noncommutative ring should have to satisfy your need?
The main motivation for me to ask this question is I am learning noncommutative algebraic geometry. In the existence work by Rosenberg, there are several kinds of spectrum(at least six different spectrum) for different purposes and they satisfy the analogue properties(noncommutative version)I mentioned above and coincide with prime spectrum when one impose the condition of commutativity. I wonder check whether these spectrum satisfied the other reasonable demand
Best Answer
I know almost nothing about noncommutative rings, but I have thought a bit about what the general concept of spectra might or should be, so I'll venture an answer.
One other property you might ask for is that it has a good categorical description. I'll explain what I mean.
The spectrum of a commutative ring can be described as follows. (I'll just describe its underlying set, not its topology or structure sheaf.) We have the category CRing of commutative rings, and the full subcategory Field of fields. Given a commutative ring $A$, we get a new category $A/$Field: an object is a field $k$ together with a homomorphism $A \to k$, and a morphism is a commutative triangle. The set of connected-components of this category $A/$Field is $\mathrm{Spec} A$.
There's a conceptual story here. Suppose we think instead about algebraic topology. Topologists (except "general" or "point-set" topologists) are keen on looking at spaces from the point of view of Euclidean space. For example, a basic thought of homotopy theory is that you probe a space by looking at the paths in it, i.e. the maps from $[0, 1]$ to it. We have the category Top of all topological spaces, and the subcategory Δ consisting of the standard topological simplices $\Delta^n$ and the various face and degeneracy maps between them. For each topological space $A$ we get a new category Δ$/A$, in which an object is a simplex in $A$ (that is, an object $\Delta^n$ of Δ together with a map $\Delta^n \to A$) and a morphism is a commutative triangle. This new category is basically the singular simplicial set of $A$, lightly disguised.
There are some differences between the two situations: the directions have been reversed (for the usual algebra/geometry duality reasons), and in the topological case, taking the set of connected-components of the category wouldn't be a vastly interesting thing to do. But the point is this: in the topological case, the category Δ$/A$ encapsulates
In the algebraic case, the category $A/$Field encapsulates
$\mathrm{Spec} A$ is the set of connected-components of this category, and so gives partial information about how $A$ looks from the point of view of fields.