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 \(\mathcal {M}\).
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Brou, P.K., Diagana, Y.M.: Indépendances affaiblies et Extensions de la Largeur analytique d’une quasi-graduation. Annales Mathématiques Africaines 4, 139–150 (2013)
Diagana, Y., Dichi, H., Sangaré, D.: Filtrations, Generalized analytic independence, Analytic spread. Afrika Matematika, Série 3 4, 101–114 (1994)
Diagana, Y.: Analytic spread of a pregraduation. In: Lecture Notes in Pure and Applied Mathematics, Commutative Ring Theory and Applications, vol. 231, pp. 107–116 (2002)
Diagana, Y.: Regular analytic independence and extensions of analytic spread. Commun. Algebra 30(6), 2745–2761 (2002)
Diagana, Y.: Quasi-graduations of rings, generalized analytic independence, extensions of the analytic spread. Afrika Matematika, Série 3 15, 93–108 (2003)
Diagana, Y.M.: Quasi-graduations of rings and modules, criteria of generalized analytic independence. Annales Mathématiques Africaines 3, 77–86 (2012)
Northcott, D.G., Rees, D.: Reductions of ideals in local rings. Proc. Camb. Philos. Soc. 50, 145–158 (1954)
Okon, J.S.: Prime divisors, analytic spread and filtrations. Pac. J. Math. 113(2), 451–462 (1984)
Valla, G.: Elementi independenti rispetto ad a ideale. Rend. Sem. Mat. Univ. Padova 44, 339–354 (1970)
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Brou, P.K., Diagana, Y.M. Quasi-graduations of modules and extensions of analytic spread. Afr. Mat. 28, 1313–1325 (2017). https://doi.org/10.1007/s13370-017-0517-5
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DOI: https://doi.org/10.1007/s13370-017-0517-5