ag.algebraic geometry - What properties "should" spectrum of noncommutative ring have?

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

how $A$ looks from the point of view of simplices.

In the algebraic case, the category $A/$Field encapsulates

how $A$ looks from the point of view of fields.

$mathrm{Spec} A$ 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.