Zariski subspace topologies on ideals

Abstract

In this paper, we show how there are tight relationships between algebraic properties of a commutative ring $R$ and topological properties of open subsets of Zariski topology on the prime spectrum of $R$. We investigate some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense and irreducible. We also give a characterization for the radical of an ideal in $R$ by using topological properties.

Keywords

• prime ideal,radical ideal,Zariski topology

References

1. [1] J. Abuhlail, A dual Zariski topology for modules, Topology Appl. 158, 457–467, 2011.
2. [2] H. Ansari-Toroghy and R. Ovlyaee-Sarmazdeh, On the prime spectrum of a module and Zariski Topologies, Comm. Algebra, 38, 4461–4475, 2010.
3. [3] M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison- Wesley, 1969.
4. [4] N. Bourbaki, Elements of Mathematics General Topology Part 1 and Part 2, Hermann and Addison-Wesley, 1966.
5. [5] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhauser, 1985.
6. [6] T. Lam, Lectures on Modules and Rings, Springer-Verlag, 1998.
7. [7] C.P. Lu, Spectra of modules, Comm. Algebra, 23, 3741–3752, 1995.
8. [8] C.P. Lu, The Zariski Topology on The Prime Spectrum of a Module, Houston J. Math. 25, 417–432, 1999 .

9. [9] C.P. Lu, Modules with Noetherian Spectrum, Comm. Algebra, 38, 807–828, 2010.
10. [10] R.L. McCasland, M.E. Moore and P.F. Smith, Rings with Noetherian spectrum, Duke Math. J. 35, 631–639, 1968.
11. [11] R.L. McCasland, M.E. Moore and P.F. Smith, On the Spectrum of a Module over a Commutative Ring, Comm. Algebra, 25, 79–103, 1997.
12. [12] D.G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University Press, 1968.
13. [13] O. Zariski and P. Samuel, Commutative Algebra Vol. 1, Springer-Verlag, 1975.