Quasi-Primary Spectrum of a Commutative Ring and a Sheaf of Rings

Abstract

In this work, the set of quasi-primary ideals of a commutative ring with identity is equipped with a topology and is called the quasi-primary spectrum. Some topological properties of this space are examined. Further, a sheaf of rings on the quasi-primary spectrum is constructed and it is shown that this sheaf is the direct image sheaf with respect to the inclusion map from the prime spectrum of a ring to the quasi-primary spectrum of the same ring.

Keywords

• Quasi-primary ideals
• Quasi-primary spectrum
• Sheaf of rings
• Zariski topology

References

1. [1] Shafarevich I. R., 2013. Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Third Edition, Springer-Verlag, Berlin.
2. [2] Ueno K., 1999. Algebraic Geometry 1: From Algebraic Varieties to Schemes, Translations of Mathematical Monographs, Vol. 185, American Mathematical Society.
3. [3] Hartshorne R., 2000. Algebraic Geometry, Springer Science+Business Media, LLC, New York.
4. [4] Özkirişci N. A., Kılıç Z., Koç S., 2018. A Note on Primary Spectrum over Commutative Rings, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), 64, pp. 111-120.
5. [5] Fuchs L., 1947. On Quasi-primary Ideals, Acta Sci. Math. (Szeged), 11(3), pp. 174-183.
6. [6] Sharp R. Y., 2001. Steps in Commutative Algebra, Second Edition, Cambridge University Press.
7. [7] Atiyah M. F., MacDonald I. G., 1969. Introduction to Commutative Algebra, Addison-Wesley Publishing Company, Inc., London.