Non-associated prime between two associated primes In the ideal has the (non-unique) primary decomposition The associated prime ideals are and is a non associated prime ideal such that A complicated example Unless for very simple examples, a primary decomposition may be hard to compute and may have a very complicated output. The following example has been designed for providing such a complicated output, and, nevertheless, being accessible to hand-written computation. Let be two homogeneous polynomials in , whose coefficients are polynomials in other indeterminates over a field . That is, and belong to and it is in this ring that a primary decomposition of the ideal is searched. For computing the primary decomposition, we suppose first that 1 is a greatest common divisor of and . This condition implies that has no primary component of height one. As is generated by two elements, this implies that it is a complete intersection (more precisely, it defines an algebraic set, which is a complete intersection), and thus all primary components have height two. Therefore, the associated primes of are exactly the primes ideals of height two that contain . It follows that is an associated prime of . Let be the homogeneous resultant in of and . As the greatest common divisor of and is a constant, the resultant is not zero, and resultant theory implies that contains all products of by a monomial in of degree . As all these monomials belong to the primary component contained in This primary component contains and , and the behavior of primary decompositions under localization shows that this primary component is In short, we have a primary component, with the very simple associated prime such all its generating sets involve all indeterminates. The other primary component contains . One may prove that if and are sufficiently generic (for example if the coefficients of and are distinct indeterminates), then there is only another primary component, which is a prime ideal, and is generated by , and . Geometric interpretation In algebraic geometry, an affine algebraic set is defined as the set of the common zeros of an ideal of a polynomial ring An irredundant primary decomposition of defines a decomposition of into a union of algebraic sets , which are irreducible, as not being the union of two smaller algebraic sets. If is the associated prime of , then and Lasker–Noether theorem shows that has