@article {
author = {Diagana, Y. M.},
title = {Quasi-bigraduations of Modules, criteria of generalized analytic independence},
journal = {Journal of Algebra and Related Topics},
volume = {6},
number = {2},
pages = {79-96},
year = {2018},
publisher = {University of Guilan},
issn = {2345-3931},
eissn = {2382-9877},
doi = {10.22124/jart.2018.11137.1113},
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 $I_{(p,q)}G_{(r,s)}\subseteqG_{(p+r,q+s)},$ 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.},
keywords = {Quasi-bigraduations,modules,generalized analytic independence},
url = {https://jart.guilan.ac.ir/article_3330.html},
eprint = {https://jart.guilan.ac.ir/article_3330_ee9644cecd6586d3400f22645cd4623f.pdf}
}