Robert Gilmer, in his paper "Commutative rings in which each prime ideal is principal", 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, 2),
$R$ 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 $R$.

I will gradually list the few things I remember here and I would appreciate if you complete it:

• Baer's criterion for injective modules 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 a ring $R$ can be alternatively defined as the projective dimension of the residue field $R/m$.

Robert Gilmer, in his paper "Commutative rings in which each prime ideal is principal", 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, 2),
$R$ 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 $R$.

I will gradually list the few things I remember here, and I would appreciate if you complete it:

• Baer's criterion for injective modules 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 $R/m$.

Robert Gilmer, in his paper "Commutative rings in which each prime ideal is principal", 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, 2),
$R$ 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 $R$.

I will gradually list the few things I remember here and I would appreciate if you complete it:

• Baer's criterion for injective modules 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 a ring $R$ can be alternatively defined as the projective dimension of the residue field $R/m$.

Robert Gilmer, in his paper "Commutative rings in which each prime ideal is principal", 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, 2),
$R$ 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 $R$.

I will gradually list the few things I remember here and I would appreciate if you complete it:

• Baer's criterion for injective modules 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 a ring $R$ can be alternatively defined as the projective dimension of the residue field $R/m$.