Editing Principal ideal (section)

==Examples and non-examples==
* The principal ideals in the (commutative) ring <math>\mathbb{Z}</math> are <math>\langle n \rangle = n\mathbb{Z}=\{\ldots,-2n,-n,0,n,2n,\ldots\}.</math> In fact, every ideal of <math>\mathbb{Z}</math> is principal (see {{section link|#Related definitions}}).
* In any ring <math>R</math>, the sets <math>\{0\}= \langle 0\rangle</math> and <math>R=\langle 1\rangle</math> are principal ideals.
* For any ring <math>R</math> and element <math>a,</math> the ideals <math>Ra,aR,</math> and <math>RaR</math> are respectively left, right, and two-sided principal ideals, by definition. For example, <math>\langle \sqrt{-3} \rangle</math> is a principal ideal of <math>\mathbb{Z}[\sqrt{-3}].</math>
* In the commutative ring <math>\mathbb{C}[x,y]</math> of [[complex number|complex]] [[polynomials]] in two [[variable (mathematics)|variable]]s, the set of polynomials that vanish everywhere on the set of points <math>\{(x,y)\in\mathbb{C}^2\mid x=0\}</math> is a principal ideal because it can be written as <math>\langle x\rangle</math> (the set of polynomials divisible by <math>x</math>).
* In the same ring <math>\mathbb{C}[x,y]</math>, the ideal <math>\langle x, y\rangle</math> generated by both <math>x</math> and <math>y</math> is ''not'' principal. (The ideal <math>\langle x, y\rangle</math> is the set of all polynomials with zero for the [[constant term]].) To see this, suppose there was a generator <math>p</math> for <math>\langle x,y\rangle,</math> so <math>\langle x, y\rangle=\langle p\rangle.</math> Then <math>\langle p\rangle</math> contains both <math>x</math> and <math>y,</math> so <math>p</math> must divide both <math>x</math> and <math>y.</math> Then <math>p</math> must be a nonzero constant polynomial. This is a contradiction since <math>p\in\langle p\rangle</math> but the only constant polynomial in <math>\langle x, y\rangle,</math> is the zero polynomial.
* In the ring <math>\mathbb{Z}[\sqrt{-3}] = \{a + b\sqrt{-3}: a, b\in \mathbb{Z} \},</math> the numbers where <math>a + b</math> is even form a non-principal ideal.  This ideal forms a regular hexagonal lattice in the complex plane.  Consider <math>(a,b) = (2,0)</math> and <math>(1,1).</math>  These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are <math>1</math> and <math>-1,</math> they are not associates.