Filtration of a ring
= Filtration of a ring
{title2=$F_iR\,F_jR\subseteq F_{i+j}R$}
A filtration of a <ring> is an exhaustive increasing family of additive subgroups $(F_iR)_{i\in\mathbb Z}$, with $1\in F_0R$ and the displayed multiplicativity condition. Its <associated graded ring> is $\bigoplus_i F_iR/F_{i-1}R$. It differs from a probabilistic <filtration>, which is a nested family of sigma-algebras.