Let ( R , m ) {\displaystyle {}(R,{\mathfrak {m}})} denote a regular local ring of dimension d {\displaystyle {}d} and of positive characteristic, let I = ( f 1 , … , f n ) {\displaystyle {}I={\left(f_{1},\ldots ,f_{n}\right)}} be an m {\displaystyle {}{\mathfrak {m}}} -primary ideal and f ∈ R {\displaystyle {}f\in R} be an element with f ∉ I {\displaystyle {}f\notin I} . Let B = R [ T 1 , … , T n ] / ( f 1 T 1 + ⋯ + f n T n + f ) {\displaystyle {}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}} be the corresponding forcing algebra.
Then the extended ideal m B {\displaystyle {}{\mathfrak {m}}B} satisfies
denote a regular local ring of dimension 
and of positive characteristic, let
be an 
-primary ideal and
be an element with
.
Let
![{\displaystyle {}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d127e2c129aac140aa8495ba705840e883347109)
be the corresponding forcing algebra.
satisfies
