University of Guilan Journal of Algebra and Related Topics 2345-3931 6 2 2018 12 01 Quasi-bigraduations of Modules, criteria of generalized analytic independence 79 96 3330 10.22124/jart.2018.11137.1113 EN Y. M. Diagana Laboratoire Math$acute{e}$matiques-Informatique, Universit$acute{e}$ Nangui Abrogoua, Abidjan, C$hat{o}$te d&#039;Ivoire Journal Article 2018 08 24 Let $\mathcal{R}$ be a ring. For a quasi-bigraduation $f=I_{(p,q)}$<br />of ${\mathcal{R}} $ \ we define an $f^{+}-$quasi-bigraduation of an ${%<br />\mathcal{R}}$-module ${\mathcal{M}}$ \ by a family $g=(G_{(m,n)})_{(m,n)\in<br />\left(\mathbb{Z}\times \mathbb{Z}\right) \cup \{\infty \}}$ of subgroups of $%<br />{\mathcal{M}}$ such that $G_{\infty }=(0) $ and $I_{(p,q)}G_{(r,s)}\subseteq<br />G_{(p+r,q+s)},$ for all $(p,q)$ and all $(r,s)\in \left(\mathbb{N} \times<br />\mathbb{N}\right) \cup \{\infty \}.$<br /> Here we show that $r$ elements of ${\mathcal{R}}$ are $J-$independent of<br />order $k$ with respect to the $f^{+}$quasi-bigraduation $g$ if and only if<br />the following two properties hold: they are $J-$independent of order $k$ with respect to the $^+$%<br />quasi-bigraduation of ring $f_2(I_{(0,0)},I)$ and there exists a relation of<br />compatibility between $g$ and $g_{I}$, where $I$ is the sub-$\mathcal{A}-$%<br />module of $\mathcal{R}$ constructed by these elements. We also show that criteria of $J-$independence of compatible<br />quasi-bigraduations of module are given in terms of isomorphisms of graded<br />algebras. https://jart.guilan.ac.ir/article_3330_ee9644cecd6586d3400f22645cd4623f.pdf