Appendix A: Mathematical Facts - Codes: The Guide to Secrecy from Ancient to Modern Times - page 483

Definition A.28 ( Permutation Groups )
The set of all permutations on the set { 1 , 2 ,...,n } is a group S n under

composition, with cardinality | S n | = n ! , called the permutation group on n

symbols .

Another notion that will help the reader understand some more advanced
concepts is the following.

Definition A.29 ( Modules)
Suppose that G is an additive abelian group, and that R is a commutative
ring with identity i that satisfy each of the following axioms:

(a) For each r ∈ R and g,h ∈ G , r ( g + h )=( rg )+( rh ) .

(b) For each r, s ∈ R and g ∈ G , ( r + s ) g =( rg )+( sg ) .

(c) For each r, s ∈ R and g ∈ G , r ( sg )=( rs ) g .

(d) For each g ∈ G , ig = g .

Then G is a ( two-sided ) module over R , or for our purposes, simply an R -
module.

Definition A.30 ( Algebras )
If R is a commutative ring with identity, then an R -algebra is a ring A such
that

(a) A is an R -module.

(b) r ( ab )=( ra ) b = a ( rb ) for all r ∈ R and a,b ∈ A .

Any R -algebra that is (as a ring) a division ring is called a division algebra .
An algebra over a field K is called a finite dimensional algebra over K .
The following notion will be needed in what follows.

Definition A.31 ( Characteristic of Rings )
The characteristic of a ring R is the smallest n ∈ N ( if there is one ) such

that n · r =0 for all r ∈ R . If there is no such n , then R is said to have

characteristic 0 .

Polynomials and Polynomial Rings
If R is a ring, then a polynomial f ( x )inan indeterminant x with coe 7 cients
in R is an infinite formal sum,

✦

f ( x )= ∞ a j x j = a 0 + a 1 x + + a n x n + ··· ··· ,

j =0