composition of the following two quotients Although the first two rings are integral domains (in fact the first is a UFD) the last is not an integral domain since it is isomorphic to showing that the ideal is not prime. (See the first property listed below.) Another non-example is the ideal since we have but neither nor are elements of the ideal. Properties An ideal in the ring (with unity) is prime if and only if the factor ring is an integral domain. In particular, a commutative ring (with unity) is an integral domain if and only if is a prime ideal. (Note that the zero ring has no prime ideals, because the ideal (0) is the whole ring.) An ideal is prime if and only if its set-theoretic complement is multiplicatively closed. Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of Krull's theorem. More generally, if is any multiplicatively closed set in , then a lemma essentially due to Krull shows that there exists an ideal of maximal with respect to being disjoint from , and moreover the ideal must be prime. This can be further generalized to noncommutative rings (see below). In the case we have Krull's theorem, and this recovers the maximal ideals of . Another prototypical m-system is the set, of all positive powers of a non-nilpotent element. The preimage of a prime ideal under a ring homomorphism is a prime ideal. The analogous fact is not always true for maximal ideals, which is one reason algebraic geometers define the spectrum of a ring to be its set of prime rather than maximal ideals; one wants a homomorphism of rings to give a map between their spectra. The set of all prime ideals (called the spectrum of a ring) contains minimal elements (called minimal prime ideals). Geometrically, these correspond to irreducible components of the spectrum. The sum of two prime ideals is not necessarily prime. For an example, consider the ring with prime ideals and (the ideals generated by and respectively). Their sum however is not prime: but its two factors are not. Alternatively, the quotient ring has zero divisors so it is not an integral domain and thus cannot be prime. Not every ideal which cannot be factored into two ideals is a prime ideal; e.g. cannot be factored but