Appendix: Review of Basic Algebra
215
Definition A.5 Let R be a set and + : R × R −→ R and · : R × R −→ R be two
binary operations on R. We say that (R, +, ·) is a ring if:
(R1) (R, +) is an abelian group. The identity element of (R, +) is called zero and
denoted by 0 R or 0 (if there is no possibility of confusion).
(R2) For all x, y, z ∈ R, x · (y · z) = (x · y) · z (associativity of ·).
(R3) For all x, y, z ∈ R, x · (y + z) = (x · y) + (x · z) and (x + y) · z = (x · z) +
(y · z) (distributivity).
If · is commutative, that is,
(R4) for all x, y ∈ G, x · y = y · x,
then the ring is called commutative.
If there exists an element 1 ∈ R such that
(R5) for every x ∈ R, 1 · x = x · 1 = x,
we say that (R, +, ·) is a ring with unit (and 1 is the unit of R).
If (R, +, ·) satisfy the conditions (R1) to (R5), then we say that (R, +, ·) is a
commutative ring with unit.
Remark A.1 The element 1 ∈ R can be also called identity or one of the ring.
The following properties are satisfied by rings.
Proposition A.4 Let (R, +, ·) be a commutative ring with unit. Then the following
are true:
(I) For all x ∈ R, 0 · x = x · 0 = 0.
(II) The unit 1 ∈ R is unique.
(III) If 1 = 0, then R consists of the single element 0.
(IV) If x, y ∈ R, then (−x) · y = x · (−y) = −(x · y) and (−x) · (−y) = x · y.
(V) If n ∈ N, and n1 = 0, then nx = 0 for all x ∈ R, where n1 = 1 + 1 + . . . + 1
n times
and N is the set of nonnegative integers.
(VI) If x, y ∈ R and n ∈ N, then one has
(x + y)
n
=
n
i=0
n
i
x
i
· y
n−i
(Binomial theorem).
Proof See the proof of Proposition 3.2 in [137].
Definition A.6 Let (R, +, ·) be a commutative ring with unit 1, where 1 = 0. We
say that (R, +, ·) is a integral domain if for all x, y ∈ R such that x · y = 0 it follows
that x = 0 or y = 0.