Associated graded ring

Remarks

The product is well defined: if $a' - a \in I^{m+1}$ and $b' - b \in I^{n+1}$, then $a'b' - ab = (a' - a) b' + a (b' - b) \in I^{m+n+1}$. With this product $\operatorname{gr}_I(R)$ is a commutative graded ring whose degree $0$ part is the quotient ring $R/I$. Example: for $R = \Z$ and $I = p\Z$ with $p$ prime, each $p^n\Z/p^{n+1}\Z$ is $\F_p$ spanned by the class of $p^n$, and $p^m p^n = p^{m+n}$, so $\operatorname{gr}_I(\Z) \cong \F_p[t]$. Non-example: the direct sum $\bigoplus_{n \ge 0} I^n$ with the same rule is not $\operatorname{gr}_I(R)$; it is the Rees algebra $R[It]$, and $\operatorname{gr}_I(R)$ is its quotient by $I R[It]$. Let $(R, \mathfrak m)$ be a Noetherian local ring with residue field $k$. Then the ring $\operatorname{gr}_{\mathfrak m}(R)$ is a polynomial ring over $k$ in $\dim R$ variables if and only if $R$ is a regular local ring.