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An associated prime of an R-module M is a prime ideal that occurs as the annihilator of some element of M. The set Ass(M) captures where M "lives" in Spec R: the zero divisors on M are exactly the union of the associated primes, the minimal associated primes correspond to the irreducible components of the support, and the embedded associated primes detect deeper non-reduced structure. For Noetherian rings, Ass(M) is finite and provides the prime-level data underlying primary decomposition.
Associated primes provide a prime-by-prime decomposition of the structure of a module. For an R-module M, a prime ideal P is an associated prime of M if P = ann(m) for some element m in M -- that is, P is the exact set of ring elements that kill some specific module element. The collection Ass(M) of all associated primes is the fundamental invariant connecting modules to the geometry of Spec R.
The most important property of associated primes is their relationship to zero divisors: in a Noetherian ring, the set of zero divisors on M equals the union of the associated primes of M. This transforms the amorphous "set of zero divisors" into a precise union of prime ideals. For a Noetherian ring R itself, the zero divisors of R are the union of Ass(R), and R is a domain if and only if Ass(R) = {(0)}. The associated primes also determine the support of M: Supp(M) is the Zariski closure of Ass(M), and the minimal primes of Supp(M) are exactly the minimal associated primes.
The distinction between minimal and embedded associated primes is geometrically significant. The minimal associated primes correspond to the irreducible components of Supp(M) -- they are the "generic points" of the locus where M lives. The embedded associated primes are strictly contained in some other associated prime and represent non-reduced or higher-order structure at special points. For example, if I = (x2, xy) in k[x, y], then R/I has associated primes (x) and (x, y). The prime (x) is minimal, corresponding to the line {x = 0}. The prime (x, y) is embedded -- the origin has "extra nilpotent structure" not captured by the minimal component. Embedded primes are not determined by the module alone in the same way minimal primes are; different primary decompositions of the same ideal can produce different embedded primes.
Associated primes connect to primary decomposition via the formula: if 0 = Q_1 ∩ ... ∩ Q_n is an irredundant primary decomposition of the zero submodule of M, then Ass(M) = {√(ann(M/Q_1)), ..., √(ann(M/Q_n))}. This gives Ass(M) as the set of primes "appearing in" the primary decomposition. The technology of associated primes extends naturally to the study of depth (the length of a maximal regular sequence in the maximal ideal, which equals the smallest i with Ext^i(R/m, M) ≠ 0 for local rings) and Cohen-Macaulay conditions, where the interplay between associated primes and regular sequences becomes central.