On submodule spectrum in multiplication le-modules
DOI:
https://doi.org/10.13069/jacodesmath.v12i2.309Keywords:
Prime submodule element, Radical element, Zariski topology, Complete lattices, Le-modulesAbstract
In this article, we have studied the Zariski topology related to a submodule element of a le-module. Obtained a base for the complement of the submodule spectrum and topological features, along with some characterizations of the radical of a submodule element, are established. Several algebraic conditions are obtained for an open subset concerning the Zariski topology to become compact, dense, Noetherian, etc.
Received: 9 November 2023 | Accepted: 13 June 2024Downloads
References
S. Ballal and V. Kharat, Zariski topology on lattice modules, Asian-Eur. J. Math. 8(4) (2015) 1550066.
S. Ballal and V. Kharat, On minimal spectrum of multiplication lattice modules, Math. Bohem. 144(1) (2019) 85–97.
M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra 4(4) (2008) 104–130.
A. K. Bhuniya and M. Kumbhakar, On irreducible pseudo-prime spectrum of topological $le$-modules, Quasigroups Related Systems 26(2) (2018) 251–262.
T. Duraivel, Topology on spectrum of modules, J. Ramanujan Math. Soc. 9(1) (1994) 25–34.
D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, Prime submodules and a sheaf on the prime spectra of modules, Comm. Algebra 42(7) (2014) 3063–3077.
M. Kumbhakar and A. K. Bhuniya, Uniqueness of primary decomposition in Laskerian $le$-module, Acta Math. Hungar. 158(1) (2019) 202–215.
M. Kumbhakar and A. K. Bhuniya, On the prime spectrum of an $le$-module, J. Algebra Appl. 20(12) (2021) 2150220.
C. P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25(3) (1999) 417–432.
C. P. Lu, A module whose prime spectrum has the surjective natural map, Houston J. Math. 33(1) (2007) 125–143.
R. McCasland, M. Moore, and P. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25(1) (1997) 79–103.
O. Öneş and M. Alkan, Multiplication modules with prime spectrum, Turk. J. Math. 43(4) (2019) 2000–2009.
O. Öneş and M. Alkan, Zariski subspace topologies on ideals, Hacet. J. Math. Stat. 48(6) (2019) 1667–1674.
N. Phadatare, S. Ballal, and V. Kharat, On the second spectrum of lattice modules, Discuss. Math. Gen. Algebra Appl. 37(1) (2017) 59–74.
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, Lecture Notes in Pure and Appl. Math., vol. 91, Dekker, New York (1984) 265–276.
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.
