and If is a minimal element of the above set, then is uniquely determined by ; in fact, is the pre-image of under the localization map . Primary ideals which correspond to non-minimal prime ideals over are in general not unique (see an example below). For the existence of the decomposition, see #Primary decomposition from associated primes below. The elements of are called the prime divisors of or the primes belonging to . In the language of module theory, as discussed below, the set is also the set of associated primes of the -module . Explicitly, that means that there exist elements in such that By a way of shortcut, some authors call an associated prime of simply an associated prime of (note this practice will conflict with the usage in the module theory). The minimal elements of are the same as the minimal prime ideals containing and are called isolated primes. The non-minimal elements, on the other hand, are called the embedded primes. In the case of the ring of integers , the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic. If an integer has prime factorization , then the primary decomposition of the ideal generated by in , is Similarly, in a unique factorization domain, if an element has a prime factorization where is a unit, then the primary decomposition of the principal ideal generated by is Examples The examples of the section are designed for illustrating some properties of primary decompositions, which may appear as surprising or counter-intuitive. All examples are ideals in a polynomial ring over a field . Intersection vs. product The primary decomposition in of the ideal is Because of the generator of degree one, is not the product of two larger ideals. A similar example is given, in two indeterminates by Primary vs. prime power In , the ideal is a primary ideal that has as associated prime. It is not a power of its associated prime. Non-uniqueness and embedded prime For every positive integer , a primary decomposition in of the ideal is The associated primes are Example: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, y2). Then M has two different minimal primary decompositions M = (y) ∩ (x, y2) = (y) ∩ (x + y, y2). The minimal prime is (y) and the embedded prime is (x, y).