Quasi-graduations of modules and extensions of analytic spread

Abstract

Let R be a ring and A a subring of R. Let $$h=\left( \mathcal {M} _{n}\right) _{n\in \mathbb {Z}\cup \left\{ +\infty \right\} }$$ be a family of subgroups of an R-module $$\mathcal {M}$$ . We say that h is an A-quasi-graduation of $$\mathcal {M}$$ if for every $$p\in \mathbb {N}, \mathcal {M}_{p}$$ is a sub-A-module of R with $$\mathcal {M}_{\infty }=(0)$$ . We present weak notions of J-independence for different extensions of the analytic spread. We show that under some conditions they coincide with $$\lim \nolimits _{n \rightarrow +\infty }\ell _{J}(h^{(n!)},A,k)$$ , where, for all integers $$p, h^{(p)} = (\mathcal {M}_{pn})_{n\in \mathbb {Z}\cup \left\{ +\infty \right\} }$$ and where $$\ell _J (h^{(p)}, A, k)$$ is the maximum number of elements of J which are J-independent of order k with respect to the A-quasi-graduation $$h^{(p)}$$ of the R-module $$\mathcal {M}$$ .