Examples

$R = C^\infty({\mathbf{R}}, {\mathbf{R}})$ , a local ring with ${\mathfrak{m}}$ the germs at zero.s
$k[x]$ and $I \coloneqq\left\langle{2, x}\right\rangle$
$k[x, y]$:
- $I\coloneqq\left\langle{x,y}\right\rangle$, the origin
- $I \coloneqq\left\langle{x, y^2}\right\rangle$, a parabola
- $I \coloneqq\left\langle{x^2, xy}\right\rangle = \left\langle{x}\right\rangle \cap\left\langle{x, y}\right\rangle^2 = \left\langle{x}\right\rangle \cap\left\langle{x^2, y}\right\rangle$
$k[x,y,z]$ and $I\coloneqq\left\langle{xy-z^2}\right\rangle$
$k[x^2, x^3]/\left\langle{x^4}\right\rangle$.
$k[x]{ {}_{ \widehat{I} } } = k {\left[\left[ x \right]\right] } $ for $I \coloneqq\left\langle{x}\right\rangle$
- This is the local ring of ${\mathbf{A}}^1_{/ {k}} $ at the origin.
$k[x,y]$ and ${\mathfrak{a}}= \left\langle{x^2 - y}\right\rangle, {\mathfrak{b}}= \left\langle{x^2+y}\right\rangle$, then ${\mathfrak{a}}+ {\mathfrak{b}}= \left\langle{x^2, y}\right\rangle$ which is not prime
A non-Noetherian ring: $M \coloneqq k + Rx$ for $R=k[x,y]$, then $M = k[\left\{{xy^k }\right\}_{k\geq 0}]$
$X = k[x,y,z]/\left\langle{xy, xz, yz}\right\rangle$ is three lines glued at the origin.

030 Examples