Robert Gilmer, in his paper "[Commutative rings in which each prime ideal is principal](https://doi.org/10.1007/BF01350233)", says:

> **Some** well known theorems indicate that certain ideal-theoretic structure properties of a commutative ring $R$ are determined by the *set of prime ideals* of $R$. For example,  
 $R$ is Noetherian if and only if each prime ideal of $R$ is finitely generated ([1](https://math.stackexchange.com/questions/1130223/if-r-is-a-commutative-ring-in-which-all-the-prime-ideals-are-finitely-generated/1130240#1130240), [2](https://math.stackexchange.com/questions/146884/an-ideal-that-is-maximal-among-non-finitely-generated-ideals-is-prime)),  
 $R$ is a multiplication ring if and only if each prime ideal of $R$ is a multiplication ideal, and    
if $R$ contains an identity, each nonzero ideal of $R$ is invertible if and only if each nonzero prime ideal of $R$ is invertible. 

I would like to have a list of  theorems that a property of a commutative ring $R$ (or $R$-modules) is determined by the prime ideals of $R$.

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I will gradually list the few things I remember here, and I would appreciate if you complete it:
- Baer's criterion for [injective modules](https://en.wikipedia.org/wiki/Injective_module) has been refined in many ways including a result that for a commutative Noetherian ring, it suffices to consider only prime ideals instead of all ideals.   
- when $R$ is a commutative Noetherian local ring with maximal ideal $m$, global dimension of the ring $R$ can be alternatively defined as the projective dimension of the residue field $R/m$.