ring of matrices over a field is a prime ideal, but it is not completely prime. This is close to the historical point of view of ideals as ideal numbers, as for the ring " is contained in " is another way of saying " divides ", and the unit ideal represents unity. Equivalent formulations of the ideal being prime include the following properties: For all and in , implies or . For any two right ideals of , implies or . For any two left ideals of , implies or . For any elements and of , if , then or . Prime ideals in commutative rings are characterized by having multiplicatively closed complements in , and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset is called an m-system if for any and in , there exists in such that is in . The following item can then be added to the list of equivalent conditions above: The complement is an m-system. Examples Any primitive ideal is prime. As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. Another fact from commutative theory echoed in noncommutative theory is that if is a nonzero -module, and is a maximal element in the poset of annihilator ideals of submodules of , then is prime. Important facts Prime avoidance lemma. If is a commutative ring, and is a subring (possibly without unity), and is a collection of ideals of with at most two members not prime, then if is not contained in any , it is also not contained in the union of . In particular, could be an ideal of . If is any m-system 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 (the primality can be proved as follows: if , then there exist elements such that by the maximal property of . We can take with . Now, if , then , which is a contradiction). In the case we have Krull's theorem, and