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.
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Notes
- 1.
A topological space X is compact if and only if every collection of closed sets in X possessing the finite intersection property, the intersection of the entire collection is non-empty (see [Chatterjee et al. (2003)]).
- 2.
Urysohn Lemma Let X be a normal space, and A and B be disjoint closed subspaces of X. Then there exists a continuous real function f defined on X, all of whose values lie in [0,1], such that f(A)=0 and f(B)=1. [Simmons 1963]
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Adhikari, M.R., Adhikari, A. (2014). Ideals of Rings: Introductory Concepts. In: Basic Modern Algebra with Applications. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1599-8_5
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DOI: https://doi.org/10.1007/978-81-322-1599-8_5
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