Editing Commutative ring (section)

=== Prime ideals ===
{{Main|Prime ideal}}
As was mentioned above, <math> \mathbb{Z} </math> is a [[unique factorization domain]]. This is not true for more general rings, as algebraists realized in the 19th century. For example, in
<math display="block">\mathbb{Z}\left[\sqrt{-5}\right]</math>
there are two genuinely distinct ways of writing 6 as a product:
<math display="block">6 = 2 \cdot 3 = \left(1 + \sqrt{-5}\right)\left(1 - \sqrt{-5}\right).</math>
Prime ideals, as opposed to prime elements, provide a way to circumvent this problem. A prime ideal is a proper (i.e., strictly contained in <math> R </math>) ideal <math> p </math> such that, whenever the product <math> ab </math> of any two ring elements <math> a </math> and <math> b </math> is in <math> p, </math> at least one of the two elements is already in <math> p .</math> (The opposite conclusion holds for any ideal, by definition.) Thus, if a prime ideal is principal, it is equivalently generated by a prime element. However, in rings such as <math>\mathbb{Z}\left[\sqrt{-5}\right],</math> prime ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any [[Dedekind ring]] (which includes <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> and more generally the [[algebraic integers|ring of integers in a number field]]) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals.

Any maximal ideal is a prime ideal or, more briefly, is prime. Moreover, an ideal <math>I</math> is prime if and only if the factor ring <math>R/I</math> is an integral domain. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. Yet another way of expressing the same is to say that the [[Complement (set theory)|complement]] <math>R \setminus p</math> is multiplicatively closed. The localisation <math>\left(R \setminus p\right)^{-1}R</math> is important enough to have its own notation: <math>R_p</math>. This ring has only one maximal ideal, namely <math>pR_p</math>. Such rings are called [[local ring|local]].