Quasi-bigraduations of Modules, criteria of generalized analytic independence

Let $mathcal{R}$ be a ring. For a quasi-bigraduation $f=I_{(p,q)}$of ${mathcal{R}} $ we define an $f^{+}-$quasi-bigraduation of an ${%mathcal{R}}$-module ${mathcal{M}}$ by a family $g=(G_{(m,n)})_{(m,n)inleft(mathbb{Z}times mathbb{Z}right) cup {infty }}$ of subgroups of $%{mathcal{M}}$ such that $G_{infty }=(0) $ and $I_{(p,q)}G_{(r,s)}subseteqG_{(p+r,q+s)},$ for all $(p,q)$ and all $(r,s)in left(mathbb{N} timesmathbb{N}right) cup {infty }.$ Here we show that $r$ elements of ${mathcal{R}}$ are $J-$independent oforder $k$ with respect to the $f^{+}$quasi-bigraduation $g$ if and only ifthe following two properties hold: they are $J-$independent of order $k$ with respect to the $^+$%quasi-bigraduation of ring $f_2(I_{(0,0)},I)$ and there exists a relation ofcompatibility between $g$ and $g_{I}$, where $I$ is the sub-$mathcal{A}-$%module of $mathcal{R}$ constructed by these elements. We also show that criteria of $J-$independence of compatiblequasi-bigraduations of module are given in terms of isomorphisms of gradedalgebras.