ABSTRACT
We define a pregraduation of a commutative ring A by a family g = (Gn ) n∈ℤ∪{+∞} of ideals of A such that G 0 = A, G ∞ = (0) and G p G q ⊆ G p+q , for all p,q ∈ ℤ. The notion of J–independence of order k with respect to a pregraduation of a ring A is defined as in [1]. We will show that r elements of G 1 are J–independent of order k with respect to a pregraduation g if and only if there exist isomorphisms from the polynomial ring with r indeterminates over https://www.w3.org/1998/Math/MathML"> A J + G k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq967.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to some https://www.w3.org/1998/Math/MathML"> A J + G k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq968.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -algebras. A weak notion of J–independence called the regular J–independence will allow to define the analytic spread of a pregraduation on a ring.
