Editing Principal ideal (section)

==Related definitions==
A ring in which every ideal is principal is called ''principal'', or a ''[[principal ideal ring]]''. A ''[[principal ideal domain]]'' (PID) is an [[integral domain]] in which every ideal is principal. Any PID is a [[unique factorization domain]]; the normal proof of unique factorization in the [[integer]]s (the so-called [[fundamental theorem of arithmetic]]) holds in any PID.

As an example, <math>\mathbb{Z}</math> is a principal ideal domain, which can be shown as follows. Suppose <math>I=\langle n_1, n_2, \ldots\rangle</math> where <math>n_1\neq 0,</math> and consider the surjective homomorphisms <math>\mathbb{Z}/\langle n_1\rangle \rightarrow \mathbb{Z}/\langle n_1, n_2\rangle \rightarrow \mathbb{Z}/\langle n_1, n_2, n_3\rangle\rightarrow \cdots.</math> Since <math>\mathbb{Z}/\langle n_1\rangle</math> is finite, for sufficiently large <math>k</math> we have <math>\mathbb{Z}/\langle n_1, n_2, \ldots, n_k\rangle = \mathbb{Z}/\langle n_1, n_2, \ldots, n_{k+1}\rangle = \cdots.</math> Thus <math>I=\langle n_1, n_2, \ldots, n_k\rangle,</math> which implies <math>I</math> is always finitely generated.  Since the ideal <math>\langle a,b\rangle</math> generated by any integers <math>a</math> and <math>b</math> is exactly <math>\langle \mathop{\mathrm{gcd}}(a,b)\rangle,</math> by induction on the number of generators it follows that <math>I</math> is principal.