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:

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

Note that $$B_IR/IB_IR = gr_IR$$, the associated graded ring to $$I$$ in $$R$$.

Particular cases
When $$I$$ is the zero ideal, $$B_IR = R$$. In other words, $$R$$ does not get blown up anywhere.

When $$I = R$$, $$B_IR$$ is the whole polynomial ring over $$R$$.