Discrete Mathematical Structures by Pearson

7-48 Discrete Mathematical Structures

7.12.2 Quotient Ring or Ring of Residue Classes

THEOREM 7.40 If S is an ideal of a ring R, then the set of R/S = {S + a, a ∈ R} of all residue

classes of S in R forms a ring with respect to the operations of addition and multiplication defined

as follows:

(S + a) + (S + b) = S + (a + b)

(S + a) (S + b) = S + ab

Proof: The set R/S is closed with respect to addition and multiplication as S + (a + b) and S + ab

are also residue classes. Both these classes are well defined as can be seen from the following:

Let S + a = S + a′ and S + b = S + b′

Then we have to show that

(S + a) + (S + b) = (S + a′) + (S + b′)

and (S + a) (S + b) = (S + a′) (S + b′)

Now S + a = S + a′⇒ a′ ∈ S + a

and S + b = S + b′⇒ b′ ∈ S + b