Download (2) (b + c) · a = b · a + c

6.6 Rings and fields
6.6.1 Rings
 Definition 21: A ring is an Abelian group [R, +] with






an additional associative binary operation
(denoted ·) such that for all a, b, cR,
(1) a · (b + c) = a · b + a · c,
(2) (b + c) · a = b · a + c · a.
We write 0R for the identity element of the group
[R, +].
For a R, we write -a for the additive inverse of a.
Remark: Observe that the addition operation is
always commutative while the multiplication need
not be.
Observe that there need not be inverses for
multiplication.
 Example: The sets Z,Q, with the usual
operations of multiplication and addition
form rings,
 [Z;+,],[Q;+,] are rings
 Let M={(aij)nn|aij is real number}, Then
[M;+,]is a ring
 Example: S,[P(S);,∩]，
 Commutative ring
Definition 23: A ring R is a commutative
ring if ab = ba for all a, bR . A ring R is
an unitary ring if there is 1R such that
1a = a1 = a for all aR. Such an
element is called a multiplicative identity.
 Example: If R is a ring, then R[x] denotes the
set of polynomials with coefficients in R. We
shall not give a formal definition of this set, but
it can be thought of as: R[x] = {a0 + a1x + a2x2
+ …+ anxn|nZ+, aiR}.
 Multiplication and addition are defined in the
usual manner; if
n
m
f ( x)   ai x and g ( x)   bi x i
i
i 0
f ( x)  g ( x) 
then
i 0
max{ n , m}
 (a  b ) x
i 0
i
i
i
nm
f ( x)  g ( x)  
k
(
a
b
)
x
 i j
k 0 i  j  k
One then has to check that these operations define a ring.
The ring is called polynomial ring.
 Theorem 6.26: Let R be a commutative
ring. Then for all a,bR,
n
(a  b) n   C (n, i )a i b n i
i 0
 where nZ+.
 1. Identity of ring and zero of ring
 Theorem 6.27: Let [R;+,*] be a ring. Then
the following results hold.
 (1)a*0=0*a=0 for aR
 (2)a*(-b)=(-a)*b=-(a*b) for a,bR
 (3)(-a)*(-b)=a*b for a,bR
 Let 1 be identity about *. Then
 (4)(-1)*a=-a for aR
 (5)(-1)*(-1)=1
 1:Identity of ring
 0:zero of ring
[M2,2(Z);+,] is an
unitary ring
a b 
M 2,2 ( Z )  {
 | a, b, c, d  Z }
c d 
 Zero of ring (0)22,
 Identity of ring is  1
1 0

  022
0 0
1

0

0  0



0 
0


1


0 0

  022
0 1
0

 0 22

1
zero - divisor of ring
2. Zero-divistors
Definition 23: If a0 is an element of a
ring R for which there exists b0 such
that ab=0(ba=0), then a(b) is called a
left(right) zero-divistor in R.
Let S={1,2}， is zero element of ring
[P(S);,∩]
6.6.2 Integral domains, division
rings and fields
 Definition 24: A commutative ring is an
integral domain if there are no zerodivisors.
 [P(S);,∩] and [M;+,] are not integral
domain, [Z;+,] is an integral domain
 Theorem 6.28: Let R be a commutative
ring. Then R is an integral domain iff. for
any a, b, cR if a0 and ab=ac, then b=c.
 Proof: 1)Suppose that R is an integral
domain. If ab = ac, then ab - ac=0
 2)R is a commutative ring, and for any a,
b, cR if a0 and ab=ac, then b=c. Prove:
R is an integral domain
 Let [R;+;*] be a ring with identity
element 1.
 If 1=0, then for aR, a=a*1=a*0=0.
 Hence R has only one element, In other
words, If |R|>1, then 10.
Definition 25: A ring is a division ring if the nonzero elements form a group under
multiplication.
If R is a division ring, then |R|2.
Ring R has identity, and any non-zero element
exists inverse element under multiplication.
Definition 26: A field is a commutative division
ring.
 [Z;+,]is a integral domain, but it is not
division ring and field
 [Q;+,], [R;+,]and[C;+,] are field
 Let [F;+,*] be a algebraic system, and |F|
2,
 (1)[F;+]is a Abelian group
 (2)[F-{0};*] is a Abelian group
 (3)For  a,b,cF, a*(b+c)=(a*b)+(a*c)
 Theorem 6.29: Any Field is an integral
domain
 Let [F;+,*] be a field. Then F is a
commutative ring.
 If a,bF-{0}, s.t. a*b =0。
 [Z;+,] is an integral domain. But it is
not a field
 Next: fields, Subring, Ideal and
Quotient ring ,Ring homomorphism
 Exercise:P381(Sixth) OR P367(Fifth) 7,8,
16,17,20
 1.Let X be any non-empty set. Show that
[P(X); ∪, ∩] is not a ring.
 2. Let Z[i] = {a + bi| a, bZ}.
 (1)Show that Z[i] is a commutative ring
and find its units.
 (2)Is Z[i] a field? Why?
 3.Show that Q[i] = {a + bi | a, bQ} is a
field.