Properties of the subtractive prime spectrum of a semimodule

Year 2023, Volume: 52 Issue: 3, 546 - 559, 30.05.2023

Song-chol Han , Won-jin Han , Won-sok Pae

https://doi.org/10.15672/hujms.1127185

Abstract

For a top semimodule over a semiring with zero and nonzero identity, this paper studies the interplay between topological properties of the subtractive prime spectrum and algebraic properties of the semimodule. We prove that the subtractive prime spectrum of the subtractively finitely generated top semimodule is a compact space, and establish necessary and sufficient conditions for the top semimodule to be subtractively finitely generated. For a multiplication semimodule over a commutative semiring, we prove that the radical of a subtractive subsemimodule coincides with its subtractive radical, that every proper subtractive subsemimodule is contained in a subtractive prime subsemimodule, that the multiplication semimodule is subtractively finitely generated iff its subtractive prime spectrum is a compact space, that in the subtractive prime spectrum, the intersection of finitely many basic open sets is compact, and that the subtractive prime spectrum of the subtractively finitely generated multiplication semimodule is a spectral space.

Keywords

Semiring, semimodule, prime subsemimodule, multiplication semimodule, Zariski topology

References

• [1] J.Y. Abuhlail, A Zariski topology for modules, Comm. Algebra 39 (11), 4163-4182, 2011.
• [2] R. Ameri, Some properties of Zariski topology of multiplication modules, Houston J. Math. 36 (2), 337-344, 2010.
• [3] R. E. Atani, Prime subsemimodules of semimodules, Int. J. Algebra 4 (26), 1299-1306, 2010.
• [4] S. E. Atani, R. E. Atani and U. Tekir, A Zariski topology for semimodules, Eur. J. Pure Appl. Math. 4 (3), 251-265, 2011.
• [5] S. E. Atani and M. S. Kohan, A note on finitely generated multiplication semimodules over commutative semirings, Int. J. Algebra 4 (8), 389-396, 2010.
• [6] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Massachusetts, 1969.
• [7] A. Barnard, Multiplication modules, J. Algebra 71, 174-178, 1981.
• [8] F. Çallialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-module M, Turkish J. Math. 41, 326-336, 2017.
• [9] Z. A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra 16 (4), 755- 779, 1988.
• [10] J. S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht, 1999.
• [11] S. C. Han, Maximal and prime k-subsemimodules in semimodules over semirings, J. Algebra Appl. 16 (7), 1750130(11 pages), 2017.
• [12] S. C. Han, k-Congruences and the Zariski topology in semirings, Hacet. J. Math. Stat. 50 (3), 699-709, 2021.
• [13] S. C. Han and W. S. Pae, Note on “Some properties of Zariski topology of multiplication modules”: Proof of compactness of basic open sets, Houston J. Math. 45 (4), 995-998, 2019.
• [14] S. C. Han, W. S. Pae and J. N. Ho, Topological properties of the prime spectrum of a semimodule, J. Algebra 566, 205-221, 2021.
• [15] U. Hebisch and H. J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, World Scientific, Singapore, 1998.
• [16] M. Henriksen, Ideals in semirings with commutative addition, Notices Amer. Math. Soc. 5, 321, 1958.
• [17] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, 43-60, 1969.
• [18] J. L. Kelley, General Topology, Springer, New York, 1975.
• [19] P. Lescot, Absolute algebra III – the saturated spectrum, J. Pure Appl. Algebra 216, 1004-1015, 2012.
• [20] C. P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25 (3), 417-432, 1999.
• [21] R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25 (1), 79-103, 1997.
• [22] O. Öneş and M. Alkan, Multiplication modules with prime spectrum, Turkish J. Math. 43, 2000-2009, 2019.
• [23] A. Peña, L. M. Ruza and J. Vielma, Separation axioms and the prime spectrum of commutative semirings, Rev. Notas Mat. 5 (2), 66-82, 2009.
• [24] Y. Tiraş, A. Harmanci and P. F. Smith, A characterization of prime submodules, J. Algebra 212, 743-752, 1999.
• [25] A. A. Tuganbaev, Multiplication modules over noncommutative rings, Sb. Math. 194 (12), 1837-1864, 2003.
• [26] G. Ulucak, U. Tekir and K. H. Oral, Separation axioms between $T_0$ and $T_1$ on lattices and lattice modules, Ital. J. Pure Appl. Math. 36, 245-256, 2016.
• [27] G. Ulucak, U. Tekir and K. P. Shum, On Spec(M) and separation axioms between $T_0$ and $T_1$, Southeast Asian Bull. Math. 39, 717-725, 2015.
• [28] H. S. Vandiver, Note on a simple type of algebra in which the cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40, 916-920, 1934.
• [29] G. Yeşilot, On prime and maximal k-subsemimodules of semimodules, Hacet. J. Math. Stat. 39, 305-312, 2010.
• [30] G. Yeşilot, On the prime spectrum of a module over noncommutative rings, Int. J. Algebra 5 (11), 523-528, 2011.
• [31] G. Yeşilot, K. H. Oral and U. Tekir, On prime subsemimodules of semimodules, Int. J. Algebra 4 (1), 53-60, 2010.
• [32] G. Zhang, Multiplication modules over any rings, J. Nanjing Univ. Math. Biquarterly 23 (1), 51-61, 2006.
• [33] G. Zhang, Properties of top modules, Int. J. Pure Appl. Math. 31 (3), 297-306, 2006.
• [34] G. Zhang and W. Tong, Spectral spaces of top right R-modules, J. Nanjing Univ. Math. Biquarterly 17 (1), 15-20, 2000.