Filtered ring

A filtered ring is a ring with subgroups $(R_n)$ satisfying $R_m{R}_n\subseteq R_{m+n}$.

A filtration of a ring is an exhaustive increasing family of additive subgroups $(F_{i}R{)}_{i\in \mathbb{Z}}$, with $1\in F_{0}R$ and the displayed multiplicativity condition. Its associated graded ring is ${⨁}_i{F}_{i}R/F_{i-1}R$. It differs from a probabilistic filtration, which is a nested family of sigma-algebras.

A negative filtration of a ring has $F_{i}R=R$ for nonnegative indices. It is complete and separated when the displayed map to the inverse limit is an isomorphism. Products make $F_{-q}R$ two-sided ideals. Series whose terms eventually belong to every $F_{-q}R$ converge. In particular, for $x\in F_{-1}R$ and $r\in R$, the geometric series ${\sum }_{j\ge 0}(xr{)}^j$ inverts $1-xr$, so the unit criterion for the Jacobson radical gives $F_{-1}R\subseteq J(R)$.

Lift finite homogeneous generators of the associated graded module of a right ideal $I$ to $x_i\in I$ of degrees $d_i$. Successive leading-symbol cancellations write an element $x\in F_{d}I$ as finite partial sums ${\sum }_i{x}_i{a}_i^{(q)}$ with residual in $F_{d-q}R$. The coefficient increments lie in $F_{d-q-d_i}R$ and therefore converge in the complete negative filtration. Multiplication is continuous, so their limits express $x$ as a finite sum ${\sum }_i{x}_i{a}_i$. This generates the actual right ideal and does not presuppose that it is closed. The left-sided statement is analogous.

For an exhaustive nonnegative increasing multiplicative filtration, each left ideal $I$ induces a homogeneous ideal $\mathrm{gr}I$ in the associated graded ring. If that graded ring is a left Noetherian ring, lift a finite homogeneous generating set of $\mathrm{gr}I$ to elements of $I$. For any element of $I$, subtract a combination of these lifts with the same leading symbol; the remainder has strictly lower degree. Induction on the nonnegative degree proves the lifts generate $I$. There is no completion hypothesis because the cancellation terminates in finitely many steps. The analogous result holds for right ideals.

For a complete separated multiplicatively filtered ring with nonnegative integer valuations, a domain associated graded ring makes the original ring a domain: the leading symbols of two nonzero elements have nonzero product. If the graded ring is left Noetherian, lift finitely many homogeneous generators of the graded ideal of a left ideal. Successively cancelling leading terms leaves remainders of increasing valuation. Completeness makes the coefficient series converge, expressing every ideal element as a finite sum of the chosen generators with completed coefficients. The right-handed argument is identical. Completeness is necessary for this infinite cancellation proof.