Primary Decomposition - Core Definitions

Primary decomposition generalizes unique prime factorization to ideals, providing a geometric decomposition into irreducible components.

DefinitionPrimary Ideal

An ideal $Q \subsetneq R$ is primary if whenever $ab \in Q$ and $a \notin Q$ , we have $b^n \in Q$ for some $n \geq 1$ .

Equivalently, $Q$ is primary if $R/Q$ has the property that every zero divisor is nilpotent. Prime ideals are primary, but not conversely.

ExamplePrimary vs. Prime

$(4) \subset \mathbb{Z}$ is primary but not prime, since $2 \cdot 2 \in (4)$ but $2 \notin (4)$ , yet $2^2 \in (4)$
$(p^n) \subset \mathbb{Z}$ is primary for any prime $p$ and $n \geq 1$
In $k[x,y]$ , the ideal $(x^2, xy)$ is not primary: $y \cdot x \in (x^2, xy)$ but $y \notin (x^2, xy)$ and no power of $x$ equals zero
$(x^2, y) \subset k[x,y]$ is primary with radical $(x, y)$

DefinitionRadical of Primary Ideal

If $Q$ is primary, its radical $\sqrt{Q} = \{a \in R : a^n \in Q \text{ for some } n\}$ is always a prime ideal, called the associated prime of $Q$ .

We say $Q$ is $\mathfrak{p}$ -primary if $\sqrt{Q} = \mathfrak{p}$ . This establishes a connection between primary ideals and prime ideals.

DefinitionPrimary Decomposition

An ideal $I \subseteq R$ has a primary decomposition if: $I = Q_1 \cap Q_2 \cap \cdots \cap Q_n$ where each $Q_i$ is primary. The decomposition is minimal or irredundant if:

No $Q_i$ is redundant (removing any $Q_i$ changes the intersection)
The radicals $\sqrt{Q_i}$ are all distinct

ExamplePrimary Decomposition in $\mathbb{Z}$

For $n = p_1^{e_1} \cdots p_k^{e_k}$ , the ideal $(n) = (p_1^{e_1}) \cap \cdots \cap (p_k^{e_k})$ is a primary decomposition, since each $(p_i^{e_i})$ is primary with associated prime $(p_i)$ .

This algebraically encodes the fundamental theorem of arithmetic.

Remark

Primary decomposition generalizes to modules: a submodule $N \subseteq M$ is primary if every zero divisor on $M/N$ is nilpotent. The theory parallels that for ideals, though with added complexity.

DefinitionAssociated Primes

For an ideal $I$ with primary decomposition $I = Q_1 \cap \cdots \cap Q_n$ , the associated primes are $\text{Ass}(R/I) = \{\mathfrak{p}_1, \ldots, \mathfrak{p}_n\}$ where $\mathfrak{p}_i = \sqrt{Q_i}$ .

The set of associated primes is independent of the choice of minimal primary decomposition, forming an intrinsic invariant of $I$ .

Remark

Geometrically, primary decomposition corresponds to decomposing an algebraic variety into irreducible components with multiplicities. The associated primes correspond to the irreducible components, while the primary ideals encode both components and multiplicities.

Primary decomposition provides an algebraic analogue of factorization, expressing ideals as intersections of "prime power" components.