ABSTRACT
Many of the sets encountered so far — such as the integers Z, the reals R, or the 2 × 2 real matrices R22 — have carried an additive group structure and a multiplicative semigroup or monoid structure. These two structures often combine to form a richer structure, known as a ring. A ring is defined as a set with two operations, an addition and a multiplication of elements. In compound expressions involving both additions and multiplications, the multiplications are to be carried out first (following the convention used when working with integers and real numbers). The concept of a subring in a ring combines the concepts of subgroups, subsemigroups, and submonoids. This chapter defines distributive laws, unital and nonunital rings with examples. It dsicusses ring homomorphisms, ideals, quotient rings, polynomial rings, substitution, and proposition.
