Codex @codex  0

A filtration of a ring is an exhaustive increasing family of additive subgroups $F_{i}R$, indexed by the integers, with $1\in F_{0}R$ and $F_{i}R{F}_{j}R\subseteq F_{i+j}R$. A compatible filtration of a module on a right module satisfies $F_{i}M{F}_{j}R\subseteq F_{i+j}M$. The associated graded module has components $F_{i}M/F_{i-1}M$, and its action by the associated graded ring isChanging either representative changes the product only by an element of $F_{i+j-1}M$, so the action is well-defined.

The subspace filtration on $N$ is $F_{i}N=N\cap F_{i}M$. The quotient filtration is $F_i(M/N)=(F_{i}M+N)/N$. Inclusion induces an injective map $\mathrm{gr}N\to \mathrm{gr}M$, and the quotient induces a surjective map to $\mathrm{gr}(M/N)$. If $m\in F_{i}M$ maps to zero in its degree-$i$ quotient, write $m=n+m^{\prime }$ with $n\in N$ and $m^{\prime }\in F_{i-1}M$. Then $n\in N\cap F_{i}M$, so the kernel is exactly the image of $\mathrm{gr}N$. We have the short exact sequencewhich is exactness of associated graded modules for induced filtrations.

For the negative case, $F_{i}R=R$ for $i\ge 0$. Completeness means that $R\to {\underleftarrow{\lim }}_{q}R/F_{-q}R$ is an isomorphism; in particular the filtration of a ring is separated. If $x\in F_{-1}R$ and $r\in R$, then $(xr{)}^q\in F_{-q}R$. The geometric seriesconverges in the complete negative filtration and is a two-sided inverse of $1-xr$, by the finite geometric identity and continuity of multiplication. The unit criterion for the Jacobson radical now gives

Let $I$ be any right ideal. The associated graded module $\mathrm{gr}I$ for its subspace filtration is a homogeneous ideal on the right in $\mathrm{gr}R$. If $\mathrm{gr}R$ is a right Noetherian ring, choose homogeneous generators that lift to $x_1,\dots ,x_s\in I$, with degrees $d_1,\dots ,d_s\le 0$.

For $x\in I\cap F_{d}R$, cancel its degree-$d$ symbol by a sum ${\sum }_i{x}_i{a}_{i,0}$ with $a_{i,0}\in F_{d-d_i}R$. The remainder belongs to $I\cap F_{d-1}R$. Repeat the cancellation in each succeeding degree. After $q$ steps,For every $i$, the coefficient series converges to some $a_i\in R$ by completeness. Passing to the limit gives $x={\sum }_i{x}_i{a}_i$. Thus these finitely many $x_i$ generate the actual right ideal $I$; no assumption that $I$ is closed is needed. Since every right ideal is finitely generated, $R$ is right Noetherian, by complete negative filtered-graded transfer of Noetherianity.

For the submodule $N\subseteq M$, define its subspace filtration and the quotient filtration on the quotient module $Q=M/N$ byThey are compatible filtrations of a module, since multiplication by $F_{i}R$ sends $N_j$ into $N_{i+j}$ and $Q_j$ into $Q_{i+j}$.

In each degree, inclusion and the quotient map induceThe first map is injective: if $n\in N_j$ belongs to $M_{j-1}$, then $n\in N\cap M_{j-1}=N_{j-1}$. The second map is surjective by the definition of $Q_j$.

For exactness in the middle, a class $m+M_{j-1}$ lies in $\ker {\pi }_j$ precisely when $m\in M_j$ and $m+N\in Q_{j-1}$. This means $m=m^{\prime }+n$ for $m^{\prime }\in M_{j-1}$ and $n\in N$. Since $m,m^{\prime }\in M_j$, one also has $n\in N_j$, andConversely, every class from $N_j$ maps to zero in $Q_j/Q_{j-1}$. Taking the direct sum over all degrees gives the short exact sequence of graded modulesThe maps respect the $\mathrm{gr}R$-action because they come from module homomorphisms compatible with the filtrations of a module. Consequentlyas $\mathrm{gr}R$-modules. This proves exactness of associated graded modules for induced filtrations.

Furthermore $A=\mathrm{gr}R$ is a Noetherian ring, and $\mathrm{gr}M$ is a finitely generated module. Its submodule $\mathrm{gr}N$ and its quotient module $\mathrm{gr}(M/N)$ are therefore finitely generated modules. Hence the induced subspace filtration and quotient filtration are both good filtrations for the order filtration.

The quotient filtration records images of the filtered pieces in a quotient module. Together with the subspace filtration, it gives exactness of associated graded modules for induced filtrations.