No Access

Zariski topology on lattice modules

Sachin Ballal

Department of Mathematics, Savitribai Phule Pune University, Pune 411 007, India

E-mail Address: ballalshyam@gmail.com

Search for more papers by this author

and

Vilas Kharat

Department of Mathematics, Savitribai Phule Pune University, Pune 411 007, India

E-mail Address: laddoo1@yahoo.com

Search for more papers by this author

https://doi.org/10.1142/S1793557115500667Cited by:2 (Source: Crossref)

Abstract

Let $M$ $M$ be a lattice module over a $C$ $C$ -lattice $L$ $L$ and $σ (M)$ $σ (M)$ be the set of all prime elements in lattice modules $M$ $M$ . In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math.52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of $σ (M)$ $σ (M)$ and algebraic properties of $M$ $M$ .

Communicated by Q. Mushtaq

Keywords:

AMSC: 06D10, 06E10, 06E99, 06F99

References

1. F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commutative ideal theory, Period. Math. Hungar. 30(1) (1995) 1–26. Crossref, Google Scholar
2. E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Damascus Univ. Basic Sci. 19 (2003) 9–20. Google Scholar
3. S. E. Atani, G. Ulucak and Ü. Tekir, The Zariski topology on semimodules, J. Adv. Res. Pure Math. 5(4) (2013) 80–90. Google Scholar
4. F. Calliap and U. Tekir, Multiplication lattice modules, Iran. J. Sci. Technol. A 4 (2011) 309–313. Google Scholar
5. D. S. Culhan, Associated primes and primal decomposition in modules and lattice modules, and their duals, Ph.D. thesis, University of California, Riverside (2005). Google Scholar
6. R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962) 481–498. Crossref, Web of Science, Google Scholar
7. E. W. Johnson and J. A. Johnson, Lattice modules over principal element domains, Comm. Algebra 7 (2003) 3505–3518. Crossref, Web of Science, Google Scholar
8. K. Keimel, A unified theory of minimal prime ideals, Acta Math. Acad. Sci. Hungar. 23 (1972) 51–69. Crossref, Web of Science, Google Scholar
9. N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276. Google Scholar
10. N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67. Web of Science, Google Scholar
11. G. Zhang, F. Wang and W. Tong, Multiplication modules in which every prime submodule is contained in a unique maximal submodule, Comm. Algebra 32(5) (2004) 1945–1959. Crossref, Web of Science, Google Scholar