1. For g2G2, (n)(N), g1G1 s.t (g1)=g2 ? surjection homomorphism.
Let : G1→G2 be a surjection homomorphism between two groups, N be a normal subgroup of G1, and KerN. Proof: G2/(N) is a group.
Theorem 6.22: Let [H;] be a normal subgroup of the group [G;]. Then [G/H;] is a group.
(N) is a subgroup of G2.
Normal subgroup?
g2-1(n)g2?(N)
For g2G2, (n)(N), g1G1 s.t (g1)=g2 ?
surjection homomorphism.

2. For g1G1, f (g1)=(N)(g1)
Let : G1→G2 be a surjection homomorphism between two groups, N be a normal subgroup of G1, and Ker N. Proof: G1/N≌G2/(N).
Corollary 6.2: If  is a homomorphism function from group [G;*] to group [G';], and it is onto, then [G/K;][G';]
Prove:f :G1→G2/(N)
For g1G1, f (g1)=(N)(g1)
1)f is a surjection homomorphism from G1 to G2/ (N).
2)Kerf =N

3. 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.

4. 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

5. 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.

6. 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
then
One then has to check that these operations define a ring.
The ring is called polynomial ring.

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

8. 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

9. 1:Identity of ring
0:zero of ring

10. [M2,2(Z);+,] is an unitary ring
Zero of ring (0)22,
Identity of ring is

11. 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);,∩]

12. 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

13. 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

14. 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.

15. 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

16. 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)

17. 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

18. 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 eR such that e*a =a for all a R.
(2)For each aR-{0}, need to find an element bR such that a*b =e.
Proof:(1)Let R={a1,a2,an}.
For cR, c 0, consider the set Rc={a1*c, a2*c, ,an*c}R.

19. Exercise:P367 7,8
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 identity.
(2)Is Z[i] a field? Why?
3.Show that Q[i] = {a + bi | a, bQ} is a field.