\(k[x,y,z]\) 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 ${\mathbb{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.