1 Definitions and basic concepts

Associated primes are a standard tool in commutative algebra for translating information about elements of a module into information about prime ideals. They are especially useful over Noetherian rings, where they interact well with annihilators, zero divisors, and decomposition theory.

1.1 Prime ideals and annihilators

Let \(R\) be a commutative ring with identity and \(M\) an \(R\)-module. For an element \(m \in M\), the annihilator of \(m\) is the ideal \[ \operatorname{Ann}_R(m) = \{r \in R : rm = 0\}. \] This ideal records which ring elements kill the chosen module element. A prime ideal is an ideal \(\mathfrak p\) such that whenever \(ab \in \mathfrak p\), then \(a \in \mathfrak p\) or \(b \in \mathfrak p\). Prime ideals are central in commutative algebra because they encode irreducible algebraic behavior.

1.2 Associated primes of a module

A prime ideal \(\mathfrak p\) is an associated prime of an \(R\)-module \(M\) if there exists a nonzero element \(m \in M\) such that \[ \mathfrak p = \operatorname{Ann}_R(m). \] Equivalently, \(\mathfrak p\) is associated to \(M\) when it is the annihilator of a cyclic submodule of \(M\) generated by a nonzero element. The set of all associated primes of \(M\) is denoted \(\operatorname{Ass}_R(M)\) or simply \(\operatorname{Ass}(M)\) when the ring is clear.

1.2.1 Associated primes of an element

An element \(m \in M\) determines the cyclic submodule \(Rm\). If \(\operatorname{Ann}_R(m)\) is prime, then \(m\) exhibits an associated prime directly. This elementwise viewpoint is often the easiest way to detect associated primes in concrete examples.

1.2.2 Associated primes of a quotient module

For a quotient module \(M/N\), associated primes are often studied through elements of the form \(m+N\). In particular, the annihilator of a class in the quotient can reflect how \(N\) sits inside \(M\). When \(M=R\) and \(N=I\) is an ideal, the associated primes of \(R/I\) are the associated primes of the quotient ring as an \(R\)-module.

1.3 Minimal and embedded associated primes

Among the associated primes of a module, those minimal with respect to inclusion are called minimal associated primes. Others, when they strictly contain a minimal associated prime, are called embedded associated primes. Minimal associated primes are closely related to the geometric or structural “largest pieces” of a module, while embedded primes often arise from hidden or repeated components in a decomposition.

2 Examples

Concrete examples help clarify the meaning of associated primes and show how they reflect divisibility and torsion phenomena.

2.1 Associated primes in principal ideal domains

Over a principal ideal domain, finitely generated modules decompose into a free part and a torsion part. The associated primes of a finitely generated torsion module are generated by prime elements appearing in its invariant factor decomposition. For example, for \[ M = \mathbb Z / 12\mathbb Z, \] the associated primes are \((2)\) and \((3)\), since \(12 = 2^2 \cdot 3\) and these prime ideals detect the prime-power torsion.

2.2 Associated primes of quotient rings

If \(R = k[x]\) and \(I = (x^n)\), then \(R/I\) has only one associated prime, namely \((x)\). More generally, for \(R = k[x,y]\) and \(I = (xy)\), the quotient \(R/I\) has associated primes \((x)\) and \((y)\). This reflects the fact that the relation \(xy=0\) creates two distinct prime directions in the module structure.

2.3 Associated primes of finitely generated modules

For a finitely generated module over a Noetherian ring, associated primes are finite in number. As a simple example, if \[ M = R/(f) \] for a nonzero element \(f\) in a unique factorization setting, the associated primes often correspond to the prime divisors of \(f\), possibly with repetitions lost in the passage to primes. Thus associated primes detect the irreducible building blocks of torsion.

3 Fundamental properties

Associated primes satisfy several basic properties that make them useful in structural arguments.

3.1 Relationship with zero divisors

An element \(r \in R\) is a zero divisor on \(M\) if there exists a nonzero \(m \in M\) such that \(rm=0\). The set of zero divisors on \(M\) is closely related to the union of associated primes: \[ Z_R(M) = \bigcup_{\mathfrak p \in \operatorname{Ass}(M)} \mathfrak p \] in the Noetherian setting. Thus associated primes precisely locate where multiplication fails to be injective on some nonzero part of the module.

3.2 Behavior under localization

Localization tends to simplify module structure while preserving the primes relevant to a chosen region of the spectrum. If \(S\) is a multiplicative subset of \(R\), then the associated primes of the localized module \(S^{-1}M\) correspond to those associated primes of \(M\) disjoint from \(S\), after extension to \(S^{-1}R\). This makes associated primes compatible with local analysis.

3.3 Behavior under extension and contraction

When passing between a ring and a localization or quotient, associated primes may extend or contract in controlled ways. A prime \(\mathfrak p\) of \(R\) can become \(S^{-1}\mathfrak p\) in \(S^{-1}R\), while primes in the localized setting contract back to primes avoiding \(S\). These operations allow associated primes to be compared across related rings.

3.4 Finiteness in Noetherian modules

If \(R\) is Noetherian and \(M\) is a Noetherian \(R\)-module, then \(\operatorname{Ass}(M)\) is finite. This finiteness is one of the most important reasons associated primes are so effective in commutative algebra. It ensures that decomposition and support arguments can be carried out with only finitely many prime ideals.

4 Primary decomposition

Primary decomposition is one of the classical settings in which associated primes appear naturally.

4.1 Primary submodules and primary ideals

A submodule \(N \subseteq M\) is called primary if whenever \(rm \in N\), then either \(m \in N\) or some power of \(r\) annihilates the image of \(m\) in \(M/N\). For ideals, this recovers the usual notion of primary ideals. Primary objects behave like “prime objects with nilpotent error,” and their radicals or radicals of annihilators are prime.

4.2 Associated primes from a primary decomposition

If a submodule or ideal admits an irredundant primary decomposition, then the radicals of the primary components yield the associated primes. In the ideal case, if \[ I = Q_1 \cap \cdots \cap Q_n \] is a primary decomposition with \(Q_i\) being \(\mathfrak p_i\)-primary, then the \(\mathfrak p_i\) are associated to \(R/I\). Embedded components in the decomposition correspond to embedded associated primes.

4.3 Uniqueness of minimal associated primes

Although primary decompositions are not unique in general, the set of minimal associated primes is intrinsic. It depends only on the module or ideal, not on the chosen decomposition. This makes minimal associated primes a robust invariant, useful for describing the irreducible support of the object.

4.4 Irredundant decompositions

In an irredundant primary decomposition, no component can be removed without changing the intersection. The associated primes arising from such a decomposition are exactly those needed to capture the module’s primary structure. Redundant components may obscure the picture, but irredundancy reveals the true prime support of the decomposition.

Associated primes are closely tied to the support of a module and to numerical invariants such as depth and dimension.

5.1 Support of a module

The support of an \(R\)-module \(M\) is the set of prime ideals \(\mathfrak p\) such that \(M_{\mathfrak p} \neq 0\). It records where the module remains nontrivial after localization. Associated primes always lie in the support, and in many Noetherian settings they identify the smallest primes in that set.

5.2 Minimal primes in the support

The minimal primes in the support often coincide with the minimal associated primes of a finitely generated module. This relationship links the geometry of the support with the algebra of annihilators. In quotient rings, minimal associated primes describe the irreducible components of the corresponding algebraic set.

5.3 Depth and associated primes

Depth measures the length of regular sequences on a module. Associated primes help detect when depth is zero: a module has depth zero at a local ring if and only if the maximal ideal is an associated prime of the localized module. Thus associated primes are a key obstruction to regular behavior.

5.4 Dimension and codimension

Associated primes often determine the dimensions of components of a module. Minimal associated primes are closely related to maximal-dimensional pieces, while embedded primes may correspond to lower-dimensional substructures. In geometric language, this distinction mirrors the difference between top-dimensional components and lower-dimensional singular or embedded parts.

6 Existence theorems

A number of standard theorems guarantee that associated primes exist under natural hypotheses.

6.1 Associated primes of Noetherian modules

Every nonzero Noetherian module over a Noetherian ring has at least one associated prime. A common proof chooses a nonzero element with maximal annihilator and shows that this annihilator must be prime. This result ensures that the theory is never empty in the Noetherian case.

6.2 Associated primes of ideals

For any ideal \(I\) in a Noetherian ring, the quotient module \(R/I\) has associated primes. These primes describe the prime ideals that occur in the algebraic structure of the ideal, especially through primary decomposition and zero-divisor analysis.

6.3 Maximal and minimal associated primes

Associated primes need not be maximal or minimal among all primes of the ring, but special cases often guarantee the existence of maximal or minimal elements within \(\operatorname{Ass}(M)\). In particular, minimal associated primes are closely linked to minimal primes over annihilators, while maximal associated primes commonly appear in local or artinian settings.

7 Operations on modules

Associated primes behave predictably under common module constructions, which makes them useful in exact and combinatorial arguments.

7.1 Submodules and quotient modules

Submodules and quotients can add or remove associated primes, depending on how they affect annihilators. The structure of a short exact sequence often controls how associated primes distribute among its terms.

7.1.1 Exact sequences

Given a short exact sequence \[ 0 \to A \to B \to C \to 0, \] the associated primes of \(B\) are constrained by those of \(A\) and \(C\). In many situations, \[ \operatorname{Ass}(A) \subseteq \operatorname{Ass}(B) \cup \operatorname{Ass}(C), \] with analogous relations for the other terms. Exactness thus provides a powerful way to track prime behavior through modules.

7.1.2 Effects on associated primes

A quotient may introduce new associated primes by collapsing relations, while a submodule may inherit some of the ambient module’s primes. However, neither operation preserves associated primes blindly; the exact module structure matters. This sensitivity is one reason associated primes are so informative.

7.2 Direct sums

For a direct sum \(M \oplus N\), the associated primes are the union of the associated primes of the summands: \[ \operatorname{Ass}(M \oplus N) = \operatorname{Ass}(M) \cup \operatorname{Ass}(N). \] This makes direct sums especially easy to analyze and provides a straightforward method for constructing modules with prescribed associated primes.

7.3 Tensor products

Tensor products can alter associated primes in more subtle ways. Under favorable hypotheses, associated primes of \(M \otimes_R N\) are related to combinations of associated primes of \(M\) and \(N\), often through support-theoretic conditions. Because tensor products interact with localization and base change, they are frequently used in geometric and homological applications of associated primes.

8 Applications

Associated primes appear throughout commutative algebra and algebraic geometry as a bridge between local and global structure.

8.1 Detecting zero divisors

A primary use of associated primes is to identify zero divisors on a module. Instead of checking all elements individually, one can study the finite set \(\operatorname{Ass}(M)\) and read off zero-divisor behavior from its union. This is particularly effective for finitely generated modules over Noetherian rings.

8.2 Studying primary decomposition

Primary decomposition becomes more transparent when interpreted through associated primes. The decomposition of ideals or submodules into primary pieces is organized by the primes attached to each component. Associated primes therefore serve as the bookkeeping device for decomposition theory.

8.3 Local algebra and localization

In local algebra, associated primes help determine which primes matter at a given local ring. They are used in studying depth, regular sequences, and local cohomological phenomena. Localization often reduces a global problem to one about a single associated prime or a small family of them.

8.4 Algebraic geometry interpretations

For quotient rings of polynomial rings, associated primes correspond to irreducible or embedded algebraic components of the underlying scheme or variety. Minimal associated primes often represent the irreducible components, while embedded primes reflect hidden lower-dimensional structure. This gives associated primes a geometric meaning in addition to their algebraic definition.

9 Variants and generalizations

The notion of associated prime has several extensions designed to work in broader contexts or to capture related phenomena.

9.1 Weakly associated primes

Weakly associated primes are primes minimal over the annihilator of some element, rather than necessarily equal to that annihilator. They can be useful when exact associated primes are too restrictive, especially outside the Noetherian setting. This variant preserves some of the elementwise intuition of the original concept.

9.2 Attached primes

Attached primes arise naturally in dual settings, especially in the study of secondary modules and local cohomology. While associated primes are connected to annihilators of elements, attached primes are tied to coassociated or secondary structure. They provide a complementary perspective on module decomposition.

9.3 Associated primes in non-Noetherian settings

In non-Noetherian rings and modules, the set of associated primes may fail to be finite or may even be empty for modules that are not zero. This makes the theory more delicate. Various weakened notions and additional hypotheses are often introduced to recover useful existence and finiteness statements.

10 See also

10.1 Prime ideal

A prime ideal is an ideal whose quotient is an integral domain and which serves as the basic geometric and algebraic building block in commutative algebra.

10.2 Primary decomposition

Primary decomposition is the expression of an ideal or submodule as an intersection of primary components, each controlled by a prime ideal.

10.3 Support of a module

The support of a module is the set of prime ideals at which the localized module is nonzero, linking module structure to the prime spectrum.