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 quasi-bigraduations of module are given in terms of isomorphisms of graded algebras.
Diagana, Y. M. (2018). Quasi-bigraduations of Modules, criteria of generalized analytic independence. Journal of Algebra and Related Topics, 6(2), 79-96. doi: 10.22124/jart.2018.11137.1113
MLA
Y. M. Diagana. "Quasi-bigraduations of Modules, criteria of generalized analytic independence". Journal of Algebra and Related Topics, 6, 2, 2018, 79-96. doi: 10.22124/jart.2018.11137.1113
HARVARD
Diagana, Y. M. (2018). 'Quasi-bigraduations of Modules, criteria of generalized analytic independence', Journal of Algebra and Related Topics, 6(2), pp. 79-96. doi: 10.22124/jart.2018.11137.1113
VANCOUVER
Diagana, Y. M. Quasi-bigraduations of Modules, criteria of generalized analytic independence. Journal of Algebra and Related Topics, 2018; 6(2): 79-96. doi: 10.22124/jart.2018.11137.1113
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