Filtered ring

A filtered ring is a ring $R$ together with an increasing family of additive subgroups $\{F_nR\}_{n\in\mathbb Z}$ (often $n\in\mathbb N$) such that:

• $F_nR\subseteq F_{n+1}R$ for all $n$,
• $1\in F_0R$,
• $F_nR\cdot F_mR \subseteq F_{n+m}R$ for all $m,n$,
• typically $\bigcup_n F_nR = R$ (exhaustive filtration).

Filtrations measure “order” or “size” of elements and produce graded approximations via the associated graded ring ; many structural arguments pass from $R$ to its graded shadow.

Examples:

• For an ideal $I\subseteq R$, the $I$-adic filtration $F_nR=I^n$ (for $n\ge 0$) is multiplicative.
• The degree filtration on $k[x_1,\dots,x_n]$ given by $F_d=\{\text{polynomials of degree}\le d\}$ is multiplicative.