复旦大学：《离散数学 Discrete Mathematics（上）》英文课件（赵一鸣）27/30

Let o: G1G2 be a surjection homomorphism between two groups, N be a normal subgroup of gi, and KerocN. Proof: G2/(N)is a group. Theorem 6.22: Let [H; be a normal subgroup of the group G; * Then G/H; o is a group. VP(N is a subgroup of g2. ◆ Normal subgroup? ◆g2op(n)g2∈?(N) ◆ For vg2∈G2,p(n)∈q(N),g1∈G1S.t q(g1)=g2 o 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

Let o: G1G2 be a surjection homomorphism between two groups, n be a normal subgroup ofG1, and Kerc∈N. Proof:G1N≌G2/@p(N) Corollary 6.2: If op is a homomorphism function from group G; to group G; and it is onto, then IG/K;lG’; ◆ Prove:f:G1→→G2/(N) ◆ For Vg1∈G12f(g1)=(N)p(g1 Ifis a surjection homomorphism from Gi to 2/@(N ◆2)kerf=N

 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

6.6 Rings and fields 6.6.1 Rings 6 Definition 21: A ring is an Abelian group r,+ with an additional associative binary operation (denoted such that for all a, b, CER, (1)a·(b+c)=a·b+a·c, (2)(b+c)·a=b·a+c·a We write oER for the identity element of the group R,+ t Fora eR. we write -a for the additive inverse of a o Remark: Observe that the addition operation is always commutative while the multiplication need not be e Observe that there need not be inverses for multiplication

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=((aiinxnlai is real number, Then M; +, x is a ring ◆ Example:S≠,P(S):,n], ◆ Commutative ring

 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, bER. A ring r is an unitary ring if there is lER such that la= al a for all aeR. Such an element is called a multiplicative identity

◆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

x Example: IfR is a ring, then rix 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: rx= a0+ajx+ a2x +…+anx叫n∈Z,a∈R} a Multiplication and addition are defined in the usual manner: if f(x)=∑a, c and g(x)=∑ bx' then max( n, m) f(x)+g(x)=∑(+b)xf(x)*g(x)=∑∑(ab,)x k=0计+j=k One then has to check that these operations define a ring The ring is called polynomial ring

 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 + a2x 2 + …+ anx n |nZ+ , aiR}.  Multiplication and addition are defined in the usual manner; if   = = = = m i i i n i i i f x a x and g x b x 0 0 ( ) ( ) then = + = + max{ , } 0 ( ) ( ) ( ) n m i i i i f x g x a b x   + = + =  = n m k k i j i j k f x g x a b x 0 ( ) ( ) ( ) One then has to check that these operations define a ring. The ring is called polynomial ring

t Theorem 6.26: Let r be a commutative ring. Then for all a, bER, (a+b)=∑C(n)ab ◆ where n∈Z+

 Theorem 6.26: Let R be a commutative ring. Then for all a,bR,  where nZ+ . = − + = n i n i n i a b C n i a b 0 ( ) ( , )

1. Identity of ring and zero of ring Theorem 6.27: Let (R; + " l be a ring. Then the following results hold (1)a*0=0*a=0 for va∈R (2)a*(-bF(-a)*b--(a b)for v a, bER ◆(3)(-a)*(-b)= a*b for va,b∈R Let I be identity about*. Then ◆(4)(-1)*a= a for va∈R ◆(5)(-1)*(-1)=1

 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 10: zero of ring

 1:Identity of ring  0:zero of ring

[M2.2(Z); +, x] is an unitary ring b M22(z) a,b,c,d∈Z} ◆ Zero of ring(0)2×2, ◆ Identity of ring is 00 ≠ ≠U2×2 10Y00 (O),x2 zero-divisor of ring 0O八01

[M2,2(Z);+,] is an unitary ring  Zero of ring (0)22 ,  Identity of ring is ( ) { | , , , } 2,2 a b c d Z c d a b M Z          =         1 1 ( )2 2 0 0 0 1 0          ( )2 2 0 0 1 0 0          ( ) 0 2 2 0 1 0 0 0 0 1 0  =                 zero - divisor of ring