# Quasi-bigraduations of Modules, criteria of generalized analytic independence

Document Type : Research Paper

Author

Laboratoire Math$acute{e}$matiques-Informatique, Universit$acute{e}$ Nangui Abrogoua, Abidjan, C$hat{o}$te d'Ivoire

Abstract

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 $g=(G_{(m,n)})_{(m,n)\in \left(\mathbb{Z}\times \mathbb{Z}\right) \cup \{\infty \}}$ of subgroups of $% {\mathcal{M}}$ such that $G_{\infty }=(0)$ and $I_{(p,q)}G_{(r,s)}\subseteq G_{(p+r,q+s)},$ for all $(p,q)$ and all $(r,s)\in \left(\mathbb{N} \times \mathbb{N}\right) \cup \{\infty \}.$
Here we show that $r$ elements of ${\mathcal{R}}$ are $J-$independent of
order $k$ with respect to the $f^{+}$quasi-bigraduation $g$ if and only if
the 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 of
compatibility 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 compatible