= Complete negative filtration
{title2=$F_iR=R\ (i\geq0),\quad R\cong\varprojlim_q R/F_{-q}R$}
A negative <filtration of a ring> has $F_iR=R$ 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> $\sum_{j\geq0}(xr)^j$ inverts $1-xr$, so the <unit criterion for the Jacobson radical> gives $F_{-1}R\subseteq J(R)$.
Back to article page
