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.

1. $Spec(R)$ is the minimal spectrum containing $Spec_{max}(R)$ which has good functoriality which means the preimage of a prime ideal is still a prime ideal.

2. if $p\epsilon Spec(R)$, $S_{p}$=$R-p$ is multiplicative set. Then one can localize.

3. $S_{p}^{-1}R$ for $p\epsilon Spec(R)$ 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

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.

1. $\text{Spec}(R)$ 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.

2. if $p\in \text{Spec}(R)$, then $S_{p} = R-p$ is multiplicative set. Then one can localize.

3. $S_{p}^{-1}R$ 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