Ideals of Rings: Introductory Concepts

Mahima Ranjan Adhikari and Avishek Adhikari
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Mahima Ranjan Adhikari: Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)
Avishek Adhikari: University of Calcutta, Department of Pure Mathematics

Chapter Chapter 5 in Basic Modern Algebra with Applications, 2014, pp 203-236 from Springer

Abstract: Abstract Chapter 5 continues the study of theory of rings, and introduces the concept of ideals which generalize many important properties of integers. Ideals and homomorphisms of rings are closely related. Like normal subgroups in the theory of groups, ideals play an analogous role in the study of rings. The real significance of ideals in a ring is that they enable us to construct other rings which are associated with the first in a natural way. Commutative rings and their ideals are closely related. Their relations develop ring theory and are applied in many areas of mathematics, such as number theory, algebraic geometry, topology, and functional analysis. In this chapter basic properties of ideals are discussed and explained with interesting examples. Ideals of rings of continuous functions and the Chinese Remainder Theorem for rings with their applications are also studied. Finally, applications of ideals to algebraic geometry Hilbert’s Nullstellensatz theorem, and the Zariski topology are discussed in this chapter.

Keywords: Introductory Concepts; Zariski Topology; Chinese Remainder Theorem; Nullstellensatz Theorem; Maximal Ideal (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-81-322-1599-8_5

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