%0 Journal Article
%T Quasi-bigraduations of Modules, criteria of generalized analytic independence
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Diagana, Y. M.
%D 2018
%\ 12/01/2018
%V 6
%N 2
%P 79-96
%! Quasi-bigraduations of Modules, criteria of generalized analytic independence
%K Quasi-bigraduations
%K modules
%K generalized analytic independence
%R 10.22124/jart.2018.11137.1113
%X 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 $I_{(p,q)}G_{(r,s)}\subseteqG_{(p+r,q+s)},$ 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 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.
%U https://jart.guilan.ac.ir/article_3330_ee9644cecd6586d3400f22645cd4623f.pdf