Rings: Introductory Concepts

Abstract

Rings also serve as a fundamental building blocks for modern algebra. Chapter 4 introduces the concept of rings, another fundamental concept in the study of modern algebra. A group is endowed with only one binary operation while a ring is endowed with two binary operations connected by some interrelations. Fields form a very important class of rings. The concept of rings arose through the attempts to prove Fermat’s last theorem and was initiated by Richard Dedekind (1831–1916) around 1880. David Hilbert (1862–1943) coined the term “ring”. Emmy Noether (1882–1935) developed the theory of rings under his guidance. A very particular but important type of rings known as commutative rings plays an important role in algebraic number theory and algebraic geometry. On the other hand, non-commutative rings are used in non-commutative geometry and quantum groups. In this chapter Wedderburn theorem on finite division rings, and some special rings, such as rings of power series, rings of polynomials, rings of continuous functions, rings of endomorphisms of abelian groups and Boolean rings are also studied.