Appendix A: Mathematical Facts - Codes: The Guide to Secrecy from Ancient to Modern Times - page 482

A.4 Groups, Fields, Modules, and Rings

Below is listed a set of axioms. Depending on which axioms are satisfied,
we are able to determine the structure of the mathematical object we wish to
define. After the listing, we describe the various types of such objects. In what
follows, denotes a set.

(a) For all α,β

S

∈ S , α + β = β + α . (Commutativity: addition)

(b) For all α,β,γ ∈ S ,( α + β )+ γ = α +( β + γ ). (Associativity: addition)

(c) such that z + α = α + z = α . (Additive

Identity) (When no confusion can arise, we use the symbol 0 here for the
additive identity z , since it mimics the ordinary zero of the integers.)

There exists a unique z ∈ S

, there exists a α (0) such that α + α (0) = α (0) + α =

(d) To each α ∈ S ∈ S

z . (Additive Inverse)

∈ S

(e) For all α,β , αβ = βα . (Commutativity: multiplication)

(f) For all α,β,γ ∈ S ,( αβ ) γ = α ( βγ ). (Associativity: multiplication)

(g) such that iα = αi = α .

(Multiplicative identity) (Here, as with the additive identity above, we can
use the symbol 1 in place of the multiplicative identity i , when no confusion
will arise from so doing, since i mimics the function of the multiplicative
identity of the integers.)

For each α ∈ S , there exists a unique i ∈ S

(h) For all α,β,γ ∈ S , α ( β + γ )= αβ + αγ . (Distributivity)

(i) For all α,β ∈ S ,if αβ = z , then α = z or β = z . (No zero divisors)

= z there exists an element denoted α − 1 such that (j) For any α ∈ T , with α

αα − 1 = i = α − 1 α .

Any set which satisfies (a)-(d) is an additive abelian group . Any set that
satisfies (a)-(d), (f), and (h) is a ring . If the ring also satisfies (e), then it
is a commutative ring . If a commutative ring also satisfies (g), then it is a
commutative ring with identity . If a ring also satisfies (i), then it is a ring with
no zero divisors . A commutative ring with identity and no zero divisors is an
integral domain , namely, those sets that satisfy all of (a)-(i). If a set satisfies
all of (a)-(j), it is a field . If a set satisfies all of (a)-(j), except (e), then it is a
skew field or division ring .
We need the following notion below. A unit or invertible element u in a
commutative ring with identity R is an element for which there exists a multi-
plicative inverse. In other words, an element u ∈ R is a unit if there exists an

element u − 1 such that uu − 1 =1 R .

The reader should note that the abstract notion of a group is any nonempty
set satisfying (b)-(d) of above. Moreover, the operation can be any binary
operation (see Definition A.4). Of particular interest in this text is the following
notion.

∈ R