Chapter 4 - QUOTIENT OR FACTOR SYSTEMS

Abstract

This chapter describes equivalence relations and partitions. A relation R on a set S is an equivalence relation if it satisfies the following conditions for arbitrary a, b, c ∈ S: (1) a R a, (2) if a R b, then b R a, and (3) if a R b and b R c, then a R c. There is a very close connection between an equivalence relation on a set S and partition. A partition of a set is a collection of nonempty disjoint subsets of which union is the whole set. For example, the subsets {1, 3, 5}, {2, 4, 6}, {7, 8} make up a partition of the set {1, 2, 3, 4, 5, 6, 7, 8}; the subset of all males and the subset of all females compose a partition of the set of all people in any country; and the subsets of parallel lines in a plane constitute a partition of the set of all lines in the plane. The chapter discusses the connection between equivalence relations and partitions is now clarified by the theorem. It discusses normal subgroups and quotient groups. The role of normal subgroups in the construction of quotient groups is played in ring theory by certain subrings called ideals. A subgroup A of the additive group of a ring R is an ideal of R if ra ∈ A and ar ∈ A, for arbitrary r ∈ R and a ∈ A. An ideal of a ring is a subring that is closed not only with respect to internal multiplication as any subring but also with respect to multiplications by arbitrary elements of the whole ring. The chapter also discusses homomorphism.