Blowup algebra

Definition

Let $R$ be a commutative unital ring and $I$ be an ideal in $R$. The blowup algebra of $I$ in $R$ is defined as:

$B_{I}R:=R\oplus I\oplus I^{2}\oplus \ldots \cong R[tI]\subseteq R[t]$

Note that $B_{I}R/IB_{I}R=gr_{I}R$, the associated graded ring to $I$ in $R$.

Particular cases

When $I$ is the zero ideal, $B_{I}R=R$. In other words, $R$ does not get blown up anywhere.

When $I=R$, $B_{I}R$ is the whole polynomial ring over $R$.