ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES

Abstract

Let $M$ be a lattice module over a $C$-lattice $L$. A proper element $P$ of $M$ is said to be classical prime if for

$a ,b\in L$ and $X\in M, abX\leq P$ implies that $aX\leq P$ or $bX\leq P$. The set of all classical prime elements of $M$, $Spec^{cp}(M)$ is called as classical prime spectrum. In this article, we introduce and study a topology on $Spec^{cp}(M)$, called as Zariski-like topology of $M$. We investigate this topological space from the point of view of spectral spaces. We show that if $M$ has ascending chain condition on classical prime radical elements, then $Spec^{cp}(M)$ with the Zariski-like topology is a spectral space.

 

Keywords

• Classical prime element,classical prime spectrum,classical prime radical element,Zariski-like topology,spectral space

References

1. F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract com- mutative ideal theory, Period. Math. Hungar., 30(1) (1995), 1-26.
2. E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Damascus Univ. J. Basic Sci., 19(2) (2003), 9-21.
3. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
4. S. Ballal and V. Kharat, Zariski topology on lattice modules, Asian-Eur. J. Math., 8(4) (2015), 1550066 (10 pp).
5. M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra, 4 (2008), 104-130.
6. M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces II, Int. Electron. J. Algebra, 4 (2008), 131-148.
7. M. Behboodi and M. J. Noori, Zariski-like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc., 35(1) (2009), 253-269.
8. N. Bourbaki, Algebre Commutative, Chap 1-2, Hermann, Paris, 1961.

9. N. Bourbaki, Elements of Mathematics, General topology, Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966.
10. F. Callialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-module M, Turkish J. Math., 41(2) (2017), 326-336.
11. K. R. Goodearl and R. B. Wareld, Jr., An Introduction to Noncommuta- tive Noetherian Rings, Second edition, London Math. Soc. Student Texts, 16, Cambridge Univ. Press, Cambridge, 2004.
12. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60.
13. J. A. Johnson, A-adic completions of Noetherian lattice modules, Fund. Math., 66 (1970), 347-373.
14. C. P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math., 25(3) (1999), 417-432.
15. C. S. Manjarekar and A. V. Bingi, Absorbing elements in lattice modules, Int. Electron. J. Algebra, 19 (2016), 58-76.
16. J. R. Munkres, Topology: a First Course, Prentice-Hall, Inc. Eglewood Clis, New Jersey, 1975.
17. N. Phadatare, S. Ballal and V. Kharat, On the second spectrum of lattice modules, Discuss. Math. Gen. Algebra Appl., 37(1) (2017), 59-74.
18. N. Phadatare and V. Kharat, On the second radical elements of lattice modules, Tbilisi Math. J., 11(4) (2018), 165-173.
19. N. Phadatare, V. Kharat and S. Ballal, On the maximal spectrum of lattice modules, Southeast Asian Bull. Math., Accepted.
20. N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, Algebra and its applications (New Delhi, 1981), Lecture Notes in Pure and Appl. Math., 91, Dekker, New York, (1984), 265-276.
21. N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. (Szeged), 52(1-2) (1988), 53-67.