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A + quasi-bigraduation of a commutative ring R is a family g = (G n,m) (n,m)∈(Z×Z)∪{+∞} of subgroups of R such that A = G 0,0 is a subring of R, G ∞ = (0) and G n,p G m,q ⊆ G n+m,p+q , for all (n, p), (m, q) ∈ (N × N). Here we show that r elements of R are J−independent of order k with respect to a + quasi-bigraduation g if and only if the two property which follow hold: they are J−independent of order k and there exists a relation of compatibility between g and g I where I is the sub-A−module of R constructed by these elements. The concept of J−independence will be extended to + quasi-bigraduations and we give criteria of J−independence in term of isomorphisms of graded algebras.

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When f = I (m,n) (m,n)∈(Z 2)∪{∞} is a + quasi-bigraduation on a ring R, a f +quasi-bigraduation of an R−module M is a family g = (G (m,n)) (m,n)∈(Z 2)∪{∞} of subgroups of M such that G ∞ = (0) and I (m,n) G (p,q) ⊆ G (m+p,n+q) , for all (m, n) and (p, q) ∈ (N×N) ∪ {∞}. We will show that r elements of R are slowly J−independent of order k with respect to a + quasi-bigraduation g on an R−module M if and only if the two property which follow hold: they are J−independent of order k with respect to the + quasi-bigraduation f 2 (A, I) and there exists a relation of compatibility between the + quasi-graduation on R deduced from g and g I where I is the sub-A−module of R constructed by these elements. We give criteria of slow J−independence of + quasi-bigraduations on an R−module in term of isomorphisms of graded algebras.