Let K {\displaystyle {}K} be a field of characteristic 0 {\displaystyle {}0} and let
Then the ideal a = ( X , Y , Z ) B {\displaystyle {}{\mathfrak {a}}=(X,Y,Z)B} has the property that H a 3 ( B ) ≠ 0 {\displaystyle {}H_{\mathfrak {a}}^{3}(B)\neq 0} . This means that in R = K [ X , Y , Z ] {\displaystyle {}R=K[X,Y,Z]} , the element X 2 Y 2 Z 2 {\displaystyle {}X^{2}Y^{2}Z^{2}} belongs to the solid closure of the ideal ( X 3 , Y 3 , Z 3 ) {\displaystyle {}{\left(X^{3},Y^{3},Z^{3}\right)}} , and hence the three-dimensional polynomial ring is not solidly closed.
be a field of characteristic 
and let
![{\displaystyle {}B=K[X,Y,Z][U,V,W]/{\left(X^{3}U+Y^{3}V+Z^{3}W-X^{2}Y^{2}Z^{2}\right)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5709a6d35131c58e75ee51473ca4590c9718ddfe)
has the property that
.
This means that in
![{\displaystyle {}R=K[X,Y,Z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db1938f9fe1b5cef4df5083228be79b93a90f72c)
,
the element 
belongs to the solid closure of the ideal 
, and hence the three-dimensional polynomial ring is not solidly closed.
