Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-30T00:56:28.953Z Has data issue: false hasContentIssue false

ON RADICAL FORMULA IN MODULES

Published online by Cambridge University Press:  01 August 2011

A. NIKSERESHT
Affiliation:
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran e-mails: a_nikseresht@shirazu.ac.ir, aazizi@shirazu.ac.ir
A. AZIZI
Affiliation:
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran e-mails: a_nikseresht@shirazu.ac.ir, aazizi@shirazu.ac.ir
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M′ such that M′ ⊕ M″ s.t.r.f of degree k, when M″ s.t.r.f of degree k. Furthermore, we will show that, under certain conditions, if the completion of a module M s.t.r.f of degree k, then there is a non-negative integer k′ such that M s.t.r.f. of degree k′. Moreover, we state a corrected version of Leung and Man's theorem (K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293) on Noetherian rings that satisfies the radical formula.

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Alkan, M. and Tiras, Y., On prime submodules, Rocky Mount. J. Math. 37 (3) (2007), 709722.CrossRefGoogle Scholar
2.Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra (Addison-Wesley, Boston, MA, 1969).Google Scholar
3.Azizi, A., Radical formula and prime submodules, J. Algebra 307 (2007), 454460.CrossRefGoogle Scholar
4.Azizi, A., Radical formula and weakly prime submodules, Glasgow Math. J. 51 (2009), 405412.CrossRefGoogle Scholar
5.Azizi, A. and Nikseresht, A., Prime bases of weakly prime submodules and the radical formula, Comm. Algebra, submitted for publication, 29 pp.Google Scholar
6.Azizi, A. and Nikseresht, A., Simplified radical formula in modules, Houston J. Math., to appear, 12 pp.Google Scholar
7.Behboodi, M. and Koohi, H., Weakly prime modules, Vietnam J. Math. 32 (2004), 185195.Google Scholar
8.Larsen, M. D. and McCarthy, P. J., Multiplicative theory of ideals (Academic Press, Oxford, UK, 1971).Google Scholar
9.Leung, K. H. and Man, S. H., On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285293.CrossRefGoogle Scholar
10.Man, S. H., On commutative Noetherian rings which have the s.p.a.r. property, Arch. Math. J. 70 (1998), 3140.CrossRefGoogle Scholar
11.Man, S. H., On commutative Noetherian rings which satisfy the generalized radical formula, Comm. Algebra 27 (8) (1999), 40754088.CrossRefGoogle Scholar
12.Matsumura, H., Commutative ring theory (Cambridge University Press, Cambridge, UK, 1992).Google Scholar
13.McCasland, R. and Moore, M., On radicals of submodules of finitely generated modules, Canad. Math. Bull. 29 (1) (1986), 3739.CrossRefGoogle Scholar
14.Parkash, A., Prime submodules and radical formulae, Contrib. Algebra. Geom. (Beiträge Algebra Geom.), to appear, 8 pp.Google Scholar
15.Pusat-Yilmaz, D. and Smith, P. F., Modules which satisfy the radical formula, Acta. Math. Hungar. 95 (2002), 155167.CrossRefGoogle Scholar
16.Sharif, H., Sharifi, Y. and Namazi, S., Rings satisfying the radical formula, Acta Math. Hungar. 71 (1996), 103108.CrossRefGoogle Scholar