Associated graded ring

Given a filtered ring $(R,F_\bullet)$ with $F_{n-1}R\subseteq F_nR$, the associated graded ring is

with multiplication induced from $R$: if $x\in F_nR$ and $y\in F_mR$, then the product of their classes in the quotients defines an element of $F_{n+m}R/F_{n+m-1}R$. Each summand is a quotient of additive groups, and the direct sum is an internal direct sum .

The ring $\mathrm{gr}_F(R)$ is naturally a graded ring ; it captures the leading-order behavior of elements of $R$ and often has simpler algebraic structure.

Examples:

• For the $I$-adic filtration, $\mathrm{gr}_I(R)=\bigoplus_{n\ge 0} I^n/I^{n+1}$.
• If the filtration is already split by degree (as in a graded ring viewed with $F_n=\bigoplus_{i\le n}R_i$), then $\mathrm{gr}_F(R)\cong R$ as graded rings.