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.  Example: The sets Z,Q, with the usual operations of multiplication and addition form rings,  [Z;+,  ],[Q;+,  ] are rings  Let M={(a ij ) n  n |a ij is real number}, Then [M;+,  ]is a ring  Example: S ,[P(S); ,∩] ，  Commutative ring

4.  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.

5.  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] = {a 0 + a 1 x + a 2 x 2 + …+ a n x n |n  Z +, a i  R}.  Multiplication and addition are defined in the usual manner; if then One then has to check that these operations define a ring. The ring is called polynomial ring.

6.  Theorem 6.26: Let R be a commutative ring. Then for all a,b  R,  where n  Z +.

7.  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

8.  1:Identity of ring  0:zero of ring

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

10. 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); ,∩]

11.  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: 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

12.  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

13.  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.

14. 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

15.  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)

16.  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

17.  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.