The procedure of identifying the prime ideals that are associated to a module or ideal: those prime ideals that occur as annihilators of some element of the module or that arise in the primary decomposition as supports of primary components.

Principle

Scan elements (or submodules) for their annihilators and collect the prime annihilators; equivalently, compute primary decompositions and read off the primes underlying the primary components. In Noetherian settings this yields a finite, well-defined set of associated primes capturing local obstructions and supports.

Demonstration

For the Z-module Z/12Z, the associated primes are (2) and (3) because 12 decomposes into 4 and 3-torsion parts and elements annihilated by 2 or 3 exhibit prime annihilators. For an ideal I in a polynomial ring, computing a primary decomposition of I and extracting the radicals of the primary ideals gives associated primes.

Misapplication

Confusing associated primes with minimal primes of the ideal of definition or assuming associated primes are always minimal; embedded associated primes can occur and are significant. Another misuse is treating associated primes computed over a localization or extension as automatically the same as over the original ring without checking change-of-ring behavior.

Consequence

Knowing associated primes clarifies the module's local structure: it identifies where torsion or embedded components live, guides localization, supports computation of depth and dimension invariants, and is essential for primary decomposition, local cohomology, and exactness tests.

Reversal

The dual viewpoint centers on minimal primes of support or on the set of primes in the support: minimal primes capture irreducible components of support while associated primes include both minimal and embedded primes, so focusing only on minimal primes collapses useful torsion information.

Boundary

Well-behaved and finite in Noetherian rings and finitely generated modules; in non-Noetherian contexts the set of associated primes may be infinite or pathological. Computation depends on having algorithms for primary decomposition or annihilator testing, and behavior can change under nonflat base change.

Semantic Tension

Semantic tension exists between associated primes and other prime-related notions (support, minimal primes, prime spectrum): associated primes detect element-level annihilators and embedded structure, while support and minimal primes emphasize geometric or irreducible decomposition. The choice depends on whether one needs elementwise torsion information or global geometric components.

Synthesis

Associated prime computation is the process of finding primes that annihilate elements of a module or appear as radicals of primary components; in Noetherian settings it yields a finite set revealing torsion and embedding phenomena, guiding localization, decomposition, and depth-related invariants, while demanding care under base change and in non-Noetherian regimes.