Algebra: Abstract and Modern

9

Rings

9.1 Examples and Elementary Properties

9.2 Certain Special Elements in Rings

9.3 The Characteristic of a Ring

9.4 Subrings

9.5 Homomorphisms of Rings

9.6 Certain Special Types of Rings

9.7 Integral Domains and Fields

It is well known that there are two familiar binary operations, namely the addition + and the multiplication · on the set ℤ of integers and that (ℤ, +) is an abelian group where as (ℤ, ·) is only a semigroup. We have earlier worked with algebraic systems, namely semigroups, monoids and groups, where there is only one binary operation in each. Now, in this chapter, we initiate the study of abstract algebraic systems having two binary operations as in the case of integers. Also, we have the rational number system, the real ...