Appendix A: Mathematical Facts - Codes: The Guide to Secrecy from Ancient to Modern Times - page 488

is a homomorphism of rings, then

R

ker( φ ) = img( φ ) .

F q is a finite field where q = p n ( p prime) and f ( x ) Example A.9 If ∈ F p [ x ]is

an irreducible, monic polynomial of degree n (see page 484), then

F q = F p [ x ]

( f ( x )) .

The situation in Example A.9 is related to the following definition and the-
orem.

Definition A.38 ( Maximal and Proper Ideals)
Let R be a commutative ring with identity. An ideal I

= R is called maximal

if whenever I ⊆ J , where J is an ideal in R , then I = J or I = R . ( An ideal

I = R is called a proper ideal. )

Theorem A.23 ( Rings Modulo Maximal Ideals).
If R is a commutative ring with identity, then M is a maximal ideal in R if
and only if R/M is a field.

Example A.10 If F is a field and r ∈ F is a fixed nonzero element, then

I = { f ( x ) ∈ F [ x ]: f ( r )=0 }

is a maximal ideal and F = F [ x ] /I .

Another aspect of rings that we will need in the text is the following. If

S = { R j : j =1 , 2 ,...,n } is a set of rings, then let R be the set of n -tuples

( r 1 ,r 2 ,...,r n ) with r j ∈ R j for j =1 , 2 ,...n , with the zero element of R being

the n -tuple, (0 , 0 ,..., 0). Define addition in R by

( r 1 ,r 2 ,...,r n )+( r 1 ,r 2 ,...,r n )=( r 1 + r 1 ,r 2 + r 2 ,...,r n + r n ) ,

for all r j ,r j ∈ R j with j =1 , 2 ,...,n , and multiplication by

( r 1 ,r 2 ,...,r n )( r 1 ,r 2 ,...,r n )=( r 1 r 1 ,r 2 r 2 ,...,r n r n ) .

This defines a structure on R called the direct sum of the rings R j , j =
1 , 2 ,...,n , denoted by

n
j =1 R j = R 1 ⊕···⊕

⊕ R n , (A.3)