a unique irredundant decomposition into irreducible algebraic varieties where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the radical of . For this reason, the primary decomposition of the radical of is sometimes called the prime decomposition of . The components of a primary decomposition (as well as of the algebraic set decomposition) corresponding to minimal primes are said isolated, and the others are said . For the decomposition of algebraic varieties, only the minimal primes are interesting, but in intersection theory, and, more generally in scheme theory, the complete primary decomposition has a geometric meaning. Primary decomposition from associated primes Nowadays, it is common to do primary decomposition of ideals and modules within the theory of associated primes. Bourbaki's influential textbook Algèbre commutative, in particular, takes this approach. Let R be a ring and M a module over it. By definition, an associated prime is a prime ideal which is the annihilator of a nonzero element of M; that is, for some (this implies ). Equivalently, a prime ideal is an associated prime of M if there is an injection of R-modules . A maximal element of the set of annihilators of nonzero elements of M can be shown to be a prime ideal and thus, when R is a Noetherian ring, there exists an associated prime of M if and only if M is nonzero. The set of associated primes of M is denoted by or . Directly from the definition, If , then . For an exact sequence , . If R is a Noetherian ring, then where refers to support. Also, the set of minimal elements of is the same as the set of minimal elements of . If M is a finitely generated module over R, then there is a finite ascending sequence of submodules such that each quotient Mi /Mi−1 is isomorphic to for some prime ideals , each of which is necessarily in the support of M. Moreover every associated prime of M occurs among the set of primes ; i.e., . (In general, these inclusions are not the equalities.) In particular, is a finite set when M is finitely generated. Let be a finitely generated module over a Noetherian ring R and N a submodule of M. Given , the set of associated primes of , there exist submodules such that and A