2. 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.

3.  1. Identity of ring and zero of ring  Theorem 6.27: Let [R;+,*] be an unitary 1 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  (4)(-1)*a=-a for  a  R  (5)(-1)*(-1)=1

4.  1:Identity of ring  0:zero of ring

5. [M 2,2 (Z);+,  ] is an unitary ring  Zero of ring (0) 2  2,  Identity of ring is

6. 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 is called a left(right) zero-divistor in R. Let S={1,2} ，  is zero element of ring [P(S); ,∩]

7.  6.6.2 Integral domains, division rings and fields  Definition 24: A commutative ring is an integral domain if there are no zero-divisors.  [P(S); ,∩] and [M;+,  ] are not integral domain, [Z;+,  ] is an integral domain  Theorem 6.28: If R is an integral domain then for any a, b, c  R if a  0 and ab=ac, then b=c.  Proof: Suppose that R is an integral domain. If ab = ac, then ab - ac=0  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.

8. Definition 25: A ring is a division ring if the non- zero 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

9.  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)

10.  Let.  Then [M 22 (Q);+,*] is a division ring. But it is not a field

11.  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

12.  Theorem 6.30: A finite integral domain is a field. integral domain :commutative, no zero-divisor Field: commutative, identity, inverse identity, inverse  Let [R;+,*] be a finite integral domain.  (1)Need to find 1  R such that 1*a =a for all a  R.  (2)For each a  R-{0}, need to find an element b  R such that a*b = 1.  Proof:(1)Let R={a 1,a 2,  a n }.  For c  R, c  0, consider the set Rc={a 1 *c, a 2 *c, ,a n *c}  R.

13.  Exercise:P367 7,8,16,17,20  1. Let Z[i] = {a + bi| a, b  Z}.  (1)Show that Z[i] is a commutative ring and find its units. Is  (2)Is Z[i] a field? Why?  2.Show that Q[i] = {a + bi | a, b  Q} is a field.