Quasi-bigraduations of Modules, criteria of generalized analytic independence

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 }=(۰) 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_۲(I_{(۰,۰)},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