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RADICAL FORMULA AND WEAKLY PRIME SUBMODULES

Published online by Cambridge University Press:  01 May 2009

A. AZIZI
A. AZIZI*
Affiliation:
Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran e-mail: [email protected]
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Abstract

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Let B be a submodule of an R-module M. The intersection of all prime (resp. weakly prime) submodules of M containing B is denoted by rad(B) (resp. wrad(B)). A generalisation of 〈E(B)〉 denoted by UE(B) of M will be introduced. The inclusions 〈E(B)〉 ⊆ UE(B) ⊆ wrad(B) ⊆ rad(B) are motivations for studying the equalities UE(B) = wrad(B) and UE(B) = rad(B) in this paper. It is proved that if R is an arithmetical ring, then UE(B) = wrad(B). In Theorem 2.5, a generalisation of the main result of [11] is given.

Keywords

13C9913C1313E0513F0513F15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 405 - 412
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Atiyah, M. F. and McDonald, I. G., Introduction to commutative algebra (Reading, MA, Adison-Wesley, 1969).Google Scholar
2.Azizi, A., On prime and weakly prime submodules, Vietnam J. Math. 36 (3) (2008), 315–325.Google Scholar
3.Azizi, A., Prime submodules and flat modules, Acta Math. Sinica Eng. Ser. 23 (1) (2007), 147–152.CrossRefGoogle Scholar
4.Azizi, A., Prime submodules of Artinian modules, Taiwanese J. Math. 13 (6) (2009).CrossRefGoogle Scholar
5.Azizi, A., Radical formula and prime submodules, J. Algebra 307 (2007), 454–460.CrossRefGoogle Scholar
6.Azizi, A., Weakly prime submodules and prime submodules, Glasgow Math. J. 48 (2) (2006), 343–346.CrossRefGoogle Scholar
7.Behboodi, M. and Koohi, H., Weakly prime submodules, Vietnam J. Math. 32 (2) (2004), 185–195.Google Scholar
8.Jenkins, J. and Smith, P. F., On the prime radical of a module over a commutative ring, Comm. Algebra 20 (12) (1992), 3593–3602.CrossRefGoogle Scholar
9.Jensen, C., Arithmetical rings, Acta Math. Sci. Hungar. 17 (1966), 115–123.CrossRefGoogle Scholar
10.Larsen, M. D. and McCarthy, P. J., Multiplicative theory of ideals (Academic, New York, 1971).Google Scholar
11.Man, S. H., One dimensional domains which satisfy the radical formula are Dedekind domains, Arch. Math. 66 (1996), 276–279.CrossRefGoogle Scholar
12.McCasland, R. L. and Moore, M. E., On radical of submodules, Comm. Algebra 19 (5) (1991), 1327–1341.CrossRefGoogle Scholar
13.Sharifi, Y., Sharif, H. and Namazi, S., Rings satisfying the radical formula, Acta Math. Hungar. 71 (1996), 103–108.CrossRefGoogle Scholar