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.