Two notes on “On the maximal spectrum of a module and Zariski topology”

Document Type : Research Paper

Authors

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

Abstract

For unitary modules over commutative rings with non-zero identity, Question 3.28 in [Bull. Malays. Math. Sci. Soc. 38(1) (2015) 303] asks if the maximal spectrum of a max-injective module is a max-spectral topological space and Proposition 3.19 in it contains an assertion that the maximal spectrum of a max-injective module is a T2-space. In this paper, we give a negative answer to the above question and show that the above assertion is false by two examples of multiplication modules.

Keywords

Main Subjects

References

1. R. Ameri, Some properties of Zariski topology of multiplication modules, Houston J. Math., (2) 36 (2010), 337-344.
2. H. Ansari-Toroghy and S. Keyvani, On the maximal spectrum of a module and Zariski topology, Bull. Malays. Math. Sci. Soc., (1) 38 (2015), 303-316.
3. Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, (4) 16 (1988), 755-779.
4. S. C. Han, W. S. Pae and J. N. Ho, Topological properties of the prime spectrum of a semimodule, J. Algebra, 566 (2021), 205-221.
5. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60.
6. R. Ovlyaee-Sarmazdeh and S. Maleki-Roudposhti, On Max-injective modules, J. Algebra Relat. Topics, (1) 1 (2013), 57-66.
Journal of Algebra and Related Topics
Volume 12, Issue 2
January 2025
Pages 47-51

Files

Share

How to cite

Statistics