Rees algebra

Remarks

It is a subring because $1 \in I^0$ and $I^m I^n \subseteq I^{m+n}$, and it is a graded ring with degree $n$ part $I^n t^n$. If $I$ is generated by $a_1, \dots, a_s$, then $R[It]$ is generated as an $R$-algebra by $a_1 t, \dots, a_s t$; so if $R$ is Noetherian, then $R[It]$ is Noetherian. Example: for $R = k[x]$, with $k$ a field, and $I = (x)$, one gets $R[It] = k[x, xt]$, and $x$ and $xt$ are algebraically independent, so this is a polynomial ring in two variables. Non-example: for a proper ideal $I$, the whole ring $R[t]$ is not $R[It]$, because $t$ lies in $R[t]$ but its coefficient $1$ does not lie in $I^1 = I$. The quotient $R[It]/I R[It]$ is the associated graded ring $\bigoplus_{n} I^n/I^{n+1}$, and the extended Rees algebra $R[It, t^{-1}]$ is used as well.