Appendix A: Mathematical Facts - Codes: The Guide to Secrecy from Ancient to Modern Times - page 487

Definition A.37 ( Ideals, Cosets, and Quotient Rings) An ideal I in a
commutative ring R with identity is a subring of R satisfying the additional
property that rI R .If I is an ideal in R then a coset of I

in R is a set of the form r + I =

⊆ I for all r ∈

{ r + α : α ∈ I } where r ∈ R . The set

{ ∈ } becomes a ring under multiplication and addition of

R/I = r + I : r R

cosets given by

( r + I )( s + I )= rs + I , and ( r + I )+( s + I )=( r + s )+ I

for any r, s R ( and this can be shown to be independent of the representatives

r and s ) . R/I is called the quotient ring of R by I , or the factor ring of R by
I , or the residue class ring modulo I . The cosets are called the residue classes
modulo I . A mapping,

∈

f : R → R/I,

which takes elements of R to their coset representatives in R/I is called the
natural map of R to R/I , and it is easily seen to be an epimorphism. The
cardinality of R/I is denoted by

| R : I | .

Example A.8 Consider the ring of integers modulo n ∈ N , Z /n Z introduced

in Definition A.20. Then n Z is an ideal in Z , and the quotient ring is the residue

class ring modulo n .

Remark A.2 Since rings are also groups, then the above concept of cosets and
quotients specializes to groups. In particular, we have the following. Note that
an index of a subgroup H in a group G can be defined similarly to the above
situation for rings as follows. The index of H in G , denoted by , is the

cardinality of the set of distinct right ( respectively, left ) cosets of H in G . Our
principal interest is when this cardinality is finite ( so this allows us to access
the definition of cardinality given earlier ) . Then Lagrange's theorem for groups
says that

| |

G : H

| G | = | G : H |·| H | ,

| |

so if G is a finite group, then | H G | . In particular, a finite abelian group G

has subgroups of all orders dividing | G | .

Now we are in a position to state the important result for rings. The reader
unfamiliar with the notation “img” of a function should consult Definition A.5
for the description.

Theorem A.22 ( Fundamental Isomorphism Theorem for Rings)
If R and S are commutative rings with identity, and

φ : R → S