1 Primary components in commutative algebra
Primary components are the building blocks that appear in decompositions of ideals into parts governed by prime ideals. They turn a global question about an ideal into a collection of “prime-specific” questions, each controlled by the behavior of powers of elements.
1.1 Prime ideals and the motivation for “primary”
The term “primary” reflects a connection to primes: each primary component typically has a prime radical, meaning the primes that matter are encoded in the component itself.
1.1.1 The role of primes in algebraic structure
In commutative algebra, prime ideals organize the geometry and arithmetic of a ring. Many invariants—such as varieties defined by ideals, or the support of modules—are described by which primes contain a given ideal. Decomposing an ideal into components indexed by primes makes these invariants easier to study, because each component focuses attention on a particular “location” in the prime spectrum.
1.1.2 Primary conditions and annihilation behavior
A primary condition is designed to capture a specific kind of “propagation” under multiplication. For an ideal \(Q\), being primary means that whenever a product \(ab\) lies in \(Q\), then either \(a\) already lies in \(Q\), or else \(b\) behaves as if it were forced into \(Q\) by taking sufficiently high powers. This mirrors how annihilation works in module settings: elements that cannot survive the action of an ideal must eventually be killed by repeated multiplication.
1.2 Primary ideals and their defining properties
Primary ideals are defined by a power-sensitive rule. Their radicals are prime, and this prime is the central datum carried by the ideal.
1.2.1 Ideals with prime radicals
For a primary ideal \(Q\), the radical \(\sqrt{Q}\) is a prime ideal. Intuitively, the radical records the “boundary” where elements cease to be nilpotent modulo \(Q\). The fact that this boundary is prime is what makes primary ideals compatible with prime-indexed decompositions.
1.2.2 Nilpotent-style control via powers
The defining property can be phrased in terms of powers: if \(ab\in Q\) and \(a\notin Q\), then there exists an exponent \(n\ge 1\) such that \(b^n\in Q\). The repeated-power requirement is what distinguishes primary ideals from general ideals. It provides a practical mechanism for tracking how multiplication interacts with containment in \(Q\).
1.3 Primary decomposition of ideals
Primary decomposition expresses an ideal as an intersection of primary ideals. The method is central for reducing structural problems to prime-labeled pieces.
1.3.1 Existence of decompositions (conceptual overview)
Under standard hypotheses (most commonly in Noetherian rings), one can decompose an ideal \(I\) as an intersection of primary ideals. Conceptually, the process isolates the contributions from different primes that appear as radicals of components. Existence results ensure that, for well-behaved rings, such a breakdown always exists, enabling systematic analysis.
1.3.2 Uniqueness up to equivalence (standard results)
While the particular primary ideals in a decomposition may vary, the decomposition is essentially unique in a controlled sense. The set of primes that occur as radicals is determined by the original ideal, and the components are unique up to the usual notion of equivalence: components with the same radical can be replaced in a way that preserves their role in the intersection. This makes primary decomposition a reliable invariant-forming tool rather than an artifact of a chosen construction.
1.3.3 Intersections of primary components
The intersection operation is the mechanism by which primary components recombine to recover the original ideal. Each component constrains the elements that must fall into the ideal when viewed near its associated prime. Intersections therefore encode “simultaneous” satisfaction of several prime-specific constraints, producing the global containment structure of \(I\).
2 Primary components of modules
For modules, the same philosophy appears in primary submodules: a module subobject can be decomposed into pieces that behave like primary ideals but in the module context.
2.1 Submodules and primary submodule concepts
Primary submodules generalize primary ideals by focusing on annihilation within a module.
2.1.1 Primary submodules versus primary ideals
A submodule \(N\subseteq M\) is called primary when it exhibits the same kind of power-sensitive absorption under multiplication by ring elements, but interpreted through the module action. If \(rm\in N\) and \(m\notin N\), then one expects some power of \(r\) to force it into the submodule. When \(M\) is the ring \(R\) itself, this notion specializes to primary ideals.
2.1.2 Associated primes of a module
Associated primes record primes that appear as annihilators of elements of \(M\) (equivalently, the primes where \(M\) has “localized” nontrivial behavior). In primary decomposition, these primes play the role analogous to radicals of primary components. Decomposing submodules typically produces components whose “support” aligns with the associated primes of the quotient module \(M/N\).
2.2 Primary decomposition of submodules
Primary decomposition of submodules refines submodule structure by representing it as an intersection of primary submodules.
2.2.1 Decomposing a submodule into intersections
A common formulation expresses \(N\) as an intersection \(N=\bigcap_i Q_i\), where each \(Q_i\) is primary in \(M\). Each \(Q_i\) captures behavior governed by a particular associated prime of \(M/N\). The decomposition reduces questions about \(N\) to questions about each \(Q_i\), which are simpler due to their power-based defining property.
2.2.2 Minimality and redundancy of components
Not all components in a decomposition contribute equally. Components can be redundant if one component contains another in a way that does not affect the intersection. Minimal primary decompositions aim to remove superfluous terms, usually ensuring that radicals (or associated primes) are represented without repetition and that no component can be omitted without changing the intersection.
2.3 Relation to localization
Localization provides a prime-by-prime viewpoint and explains why primary components reflect local behavior.
2.3.1 Local viewpoints at primes
When localizing at a prime \(\mathfrak p\), parts of the module supported away from \(\mathfrak p\) often become invisible. Primary components whose associated prime differs from \(\mathfrak p\) tend to behave in a way that either becomes redundant or simplifies drastically. This creates a direct correspondence between decompositions and the localized structure near each prime.
2.3.2 How components reflect local behavior
Each primary component can be viewed as governing a neighborhood of its associated prime: after localization, one expects the decomposition to collapse to the component(s) relevant to that prime. In this way, primary components act like “local controllers” of containment and annihilation, turning global decompositions into a collection of localized truths.
3 Operations and properties
Primary components interact with standard algebraic operations. Their radicals, containment relations, and functorial behavior help organize how decompositions behave under changes to ideals and modules.
3.1 Intersections, sums, and containment relations
Because decompositions are built from intersections, it is natural to ask how primary components behave under inclusion and how they recombine with sums.
3.1.1 How primary components behave under inclusion
If one compares two decompositions or two ideals/submodules, inclusion relations often propagate through radicals and through component containment. A smaller ideal typically forces more constraints and can yield components with radicals that refine or enlarge in a controlled manner. However, inclusion of components is not merely monotone in an obvious way; rather, the relationship is regulated by how radicals and annihilators interact.
3.1.2 Component behavior under sums and intersections
Intersections are directly compatible with primary decompositions: intersecting ideals corresponds to imposing constraints together, which can sometimes be expressed using refinements of decompositions. Sums are more subtle because they correspond to “weakening” constraints, and the primary decomposition of a sum may not be obtained by a simple operation on the decompositions of summands. Still, understanding radicals and associated primes of the resulting object often guides the computation.
3.2 Radicals and prime correspondences
Radicals are the bridge between primary components and prime ideals, enabling correspondences between decomposition data and prime-related invariants.
3.2.1 The prime radical of a primary component
Every primary component carries a prime radical: for an ideal \(Q\), \(\sqrt{Q}\) is prime; for a primary submodule, an analogous prime appears through annihilation considerations in the quotient. This prime is central because it identifies the associated location where the component governs the structure.
3.2.2 Tracking associated primes through decomposition
As components are refined or made minimal, the set of primes occurring as radicals corresponds to the set of associated primes of the quotient by the decomposed subobject. This tracking is a common strategy: rather than reconstructing the entire decomposition, one can often deduce which primes must appear by examining associated primes.
3.3 Functorial aspects
Primary decomposition behaves predictably under algebraic maps and constructions, reflecting how module-theoretic properties transfer.
3.3.1 Effects of module homomorphisms
Under a module homomorphism, submodules map to submodules, and preimages preserve intersection patterns. If a decomposition exists for a submodule, then applying suitable functorial operations (such as taking inverse images or induced maps on quotients) can transfer the decomposition structure to related subobjects. The precise form depends on injectivity, surjectivity, and whether localization or completion is involved.
3.3.2 Stability under standard algebraic constructions
Operations such as taking quotients, passing to submodules, and localizing commonly preserve the “primary nature” of relevant components, though some components may vanish or merge. In Noetherian contexts, these transformations usually maintain the existence of primary decompositions, while altering the component data in ways controlled by associated primes.
4 Examples and computations (algebra-focused)
Concrete computations illustrate how primary decomposition turns abstract definitions into checkable containment and power conditions.
4.1 Simple ideal decompositions
Elementary decompositions are often products of prime ideals or intersections derived from them.
4.1.1 Decomposing products into primary pieces (conceptual)
A guiding pattern is that ideals built from prime data often decompose into intersections of primary ideals. For example, if an ideal resembles a product of powers of primes, its decomposition can frequently be organized so that each component reflects one prime factor. Even when explicit formulas are not immediate, these examples show how multiplication by prime powers leads naturally to primary behavior.
4.1.2 Computing radicals for components
To identify the prime associated with a component, one computes the radical of the candidate primary ideal. In examples, radicals frequently reveal which primes are “active” in the decomposition. This step is crucial: once \(\sqrt{Q}\) is known to be prime, \(Q\) is expected to fit into the primary framework.
4.2 Module decomposition examples
Module settings often reduce to quotient computations and annihilator analysis.
4.2.1 Primary decomposition in quotient modules
When studying a submodule \(N\subseteq M\), one can shift attention to the quotient \(M/N\). Associated primes and annihilators in the quotient indicate which primary submodules should appear in a decomposition of \(N\). In practice, constructing components often proceeds by selecting submodules whose quotients have controlled annihilator behavior aligned with the relevant primes.
4.2.2 Using annihilators to identify components
Annihilators of elements (or of subquotients) help locate prime radicals tied to primary pieces. By finding elements whose annihilators correspond to a given prime, one can guess which primary component must account for that prime. This approach connects the operational definition of primary submodules with the computable invariant of annihilator ideals.
4.3 Practical notes for calculations
Computational methods emphasize how to choose candidates for components and how to verify the defining primary condition.
4.3.1 Choosing generators for components
In explicit rings, candidate primary ideals or primary submodules are often proposed using generators that reflect expected prime radicals, such as powers of elements or ideals corresponding to prime factors. Once a candidate is selected, one checks whether it fits the primary rule and whether its intersection with other candidates reproduces the target subobject.
4.3.2 Verifying primary conditions computationally
To verify that \(Q\) is primary, one checks the power condition: from inclusions \(ab\in Q\), test whether failure of \(a\in Q\) forces some \(b^n\in Q\). Computationally, this is typically done by reasoning about radicals and nilpotence modulo \(Q\), since the radical being prime is a strong constraint. In module computations, analogous checks translate to containment conditions on elements acting on \(M/N\) and to verifying that annihilation occurs through powers.