Complete intersection ring

Regular local rings are complete intersection rings, but the converse is not true: the ring ${\displaystyle k[x]/(x^{2})}$ is a 0-dimensional complete intersection ring that is not regular.

An example of a locally complete intersection ring which is not a complete intersection ring is given by ${\displaystyle k[x,y]/(y-x^{2},x^{3})}$which has length 3 since it is isomorphic as a ${\displaystyle k}$ vector space to ${\displaystyle k\oplus k\cdot x\oplus k\cdot x^{2}}$.[1]

Complete intersection local rings are Gorenstein rings, but the converse is not true: the ring ${\displaystyle k[x,y,z]/(x^{2},y^{2},xz,yz,z^{2}-xy)=R/I}$ is a 0-dimensional Gorenstein ring that is not a complete intersection ring. As a ${\displaystyle k}$-vector space this ring is isomorphic to

${\displaystyle {\frac {k[x,y,z]}{(x^{2},y^{2},xz,yz,z^{2}-xy)}}\cong R_{0}\oplus R_{1}\oplus R_{2}}$, where ${\displaystyle R_{0}=k\cdot 1,R_{1}=k\cdot x\oplus k\cdot y\oplus k\cdot z}$, and ${\displaystyle R_{2}=k\cdot z^{2}}$

showing it is Gorenstein since the top-degree component is dimension ${\displaystyle 1}$ and it satisfies the Poincare property. It is not a local complete intersection ring because the ideal ${\displaystyle I\subset R}$ is not ${\displaystyle R}$-regular. For example, ${\displaystyle xy}$ is a zero-divisor to ${\displaystyle x}$ in ${\displaystyle R/(x^{2},y^{2})}$.