文档介绍：Computer Science and Information Engineering
National Chi Nan University
Combinatorial Mathematics
Dr. Justie Su-Tzu Juan
Chapter 14 Rings and Modular
Arithmetic
§ The Ring Structure: Definition and
Examples
Slides for a Course Based on the Text
Discrete & Combinatorial Mathematics (5th Edition)
by Ralph P. Grimaldi
(c) Spring 2007, Justie Su-Tzu Juan 1
§ The Ring Structure: Definition and Examples
Def : (R, , ) is a ring, where
R: a nonempty set
: R R R, :R R R: two closed binary operations
and a, b, c R satisfied:
a) a b = b mutative Law of
b) a (b c) = (a b) c Associative Law of
c) z R . a z = z a = a a R Existence of an identity for
d) a R, a R . a b = b a = z Existence of inverses under
e) a (b c) = (a b) c Associative Law of
f) a (b c) = (a b) (a c) Distributive Laws of over
(b c) a = (b a) (c a)
(c) Spring 2007, Justie Su-Tzu Juan 2
§ The Ring Structure: Definition and Examples
Note: ring (R, +, )
a + b + c = (a + b) + c = a + (b + c)
a b c = (a b) c = a (b c)
a b ab
r, n Z+ with n 3 and 1 r < n:
(a1 + a2 + …+ ar) + (ar+1 + …+ an) = a1 + a2 + …+ ar + ar+1 + …+ an
(a1 a2 …ar) (ar+1 …an) = a1 a2 …ar ar+1 …an
n Z+, where n 3:
a(b1 + b2 + …+ bn) = ab1 + ab2 + …+ abn
(b1 + b2 + …+ bn)a = b1a + b2a + …+ bna
Ex : +: ordinary addition, : ordinary multiplication:
(Z, + , ), (R, +, ), (Q, +, ), (C, +, ) are rings.
additive identity z integer 0
additive inverse of x x
(c) Spring 2007, Justie Su-Tzu Juan 3
§ The Ring Structure: Definition and Examples
Ex : Let M2(Z) denote the set of all 2 2 matrixes
. M ( ) = a b
2 Z : a,b,c,d Z
c d
(M2(Q), M2(r), M2(C) are defined similarly)
define: a be f
a e, b f , c g, d h
c dg h
e f a e b f
a b
c dg hc g d h
e f ae bg af b