the ring is a 0-dimensional complete intersection ring that is not regular. Not a complete intersection An example of a locally complete intersection ring which is not a complete intersection ring is given by which has length 3 since it is isomorphic as a vector space to . Counterexample Complete intersection local rings are Gorenstein rings, but the converse is not true: the ring is a 0-dimensional Gorenstein ring that is not a complete intersection ring. As a -vector space this ring is isomorphic to , where , and showing it is Gorenstein since the top-degree component is dimension and it satisfies the Poincare property. It is not a local complete intersection ring because the ideal is not -regular. For example, is a zero-divisor to in . Citations References Commutative algebra