Forcing algebra/Relation to closure operations/description

Definition

Let ${}R$ be a commutative ring and let ${}f_{1},\ldots ,f_{n}$ and ${}f$ be elements in ${}R$. Then the ${}R$-algebra

$R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}$

is called the forcing algebra of these elements (or these data).

This algebra was introduced by Hochster. The forcing algebra forces that ${}f$ belongs to the extended ideal ${}(f_{1},\ldots ,f_{n})B$. It yields a scheme morphism

$\varphi \colon {\rm {Spec}}\,B\longrightarrow X={\rm {Spec}}\,R.$

We are interested in the relationship:

How is ${}f$ related to ${}I$?

Does ${}f$ belong to certain closure operations of ${}I$? ${}\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Longleftrightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ Properties of ${}\varphi$.

Examples

$f\in I\Longleftrightarrow \varphi {\text{ has a scheme-section}}.$

$f\in {\rm {rad}}\,I\Longleftrightarrow \varphi {\text{ is surjective}}.$

$f\in {\overline {I}}\,\,({\text{integral closure}})\Longleftrightarrow \varphi {\text{ is a universal submersion}}.$