QUASI-BIGRADUATIONS OF MODULES, CRITERIA OF GENERALIZED ANALYTIC INDEPENDENCE

Let R be a ring. For a quasi-bigraduation f = I(p,q) of R we define an f +−quasi-bigraduation of an R-module M by a family g = (G(m,n))(m,n)∈(Z×Z)∪{∞} of subgroups of M such that G∞ = (0) and I(p,q)G(r,s) ⊆ G(p+r,q+s) , for all (p, q) and all (r, s) ∈ (N × N) ∪ {∞}. Here we show that r elements of 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 f2(I(0,0), I) and there exists a relation of compatibility between g and gI , where I is the sub-A−module of R constructed by these elements. We also show that criteria of J−independence of compatible quasibigraduations of module are given in terms of isomorphisms of graded algebras.