复旦大学：《离散数学 Discrete Mathematics（上）》英文课件（赵一鸣）29/30

O 6.6.4 Subring, Ideal and Quotient ring Subring Definition 29: A subring of a ring r is a s nonempty subset s of r which is also a ring under the same operations Example: [OV2; + x]is a subring of [R; t ], where R is real number set

6.6.4 Subring, Ideal and Quotient ring 1. Subring Definition 29: A subring of a ring R is a nonempty subset S of R which is also a ring under the same operations. Example : [ Q 2;+,]is a subring of [ R;+,] , where R i s real number set

Theorem 6.34: A subset s of a ring r is a subring if and only if for a, bEs: Da+bEs (2)-a∈S 3a bES

 Theorem 6.34: A subset S of a ring R is a subring if and only if for a, bS：  (1)a+bS  (2)-aS  (3)a·bS

Example: Let R; +, be a ring. Then C={xx∈R, and ax= xa for all a∈R}isa subring of r Proof: Forex,y∈C,x+y,-X∈?C,xy?∈Ci,e VaER, a (x+y=?(x+y) a, a (-x=? KX)'a a'(x'y)=?(x'y).a

Example: Let [R;+,·] be a ring. Then C={x|xR, and a·x=x·a for all aR} is a subring of R. Proof: For  x,yC, x+y,-x?C, x·y?C i.e.  aR,a·(x+y)=?(x+y)·a,a·(-x)=?(- x)·a,a·(x·y) =?(x·y)·a

Q2de(浬想) Definition 30. Let R;+,* be a ring. A subring s of r is called an ideal of r ifrs ∈ S and sres for any∈ Rand se∈S t To show that s is an ideal of r it is sufficient to check that .(a)s;+ is a subgroup of r;+; ◆(b)ifre∈ R ses, then rses and sres

 2.Ideal(理想)  Definition 30:. Let [R; + , * ] be a ring. A subring S of R is called an ideal of R if rs S and srS for any rR and sS.  To show that S is an ideal of R it is sufficient to check that  (a) [S; +] is a subgroup of [R; + ];  (b) if rR and sS, then rsS and srS

/ Example: IR; + *I is a commutative ring with identity element. For aER, (a=fa*rrer, then (a);, I is an ideal of [R; +, 5. If(R; + is a commutative ring, For Va ∈R,(a)={ar+nar∈R,n∈},then l(a);t, is an ideal of r; +

Example: [R;+,*] is a commutative ring with identity element. For aR， (a)={a*r|rR},then [(a);+,*] is an ideal of [R;+,*].  If [R;+,*] is a commutative ring, For a R, (a)={a*r+na|rR,nZ}, then [(a);+,*] is an ideal of [R;+,*]

◆ Principle ideas Definition 31: If R is a commutative ring and a∈R,then(a)={a*r+na|r∈R} is the principle ideal defined generated by a. Example: Every ideal in Z; +, is a principle. t Proof: Let d be an ideal of z If D=0, then it holds ◆ Suppose that D≠{0}. ◆Letb= maed{a||a≠0, where a∈D}

Principle ideas  Definition 31: If R is a commutative ring and aR, then (a) ={a*r+na|rR} is the principle ideal defined generated by a.  Example: Every ideal in [Z;+,*] is a principle.  Proof: Let D be an ideal of Z.  If D={0}, then it holds.  Suppose that D{0}.  Let b=minaD{|a| | a0,where a D}

3. Quotient ring Theorem 6.35: Let R;+, be a ring and let s be an ideal of r Ifr/S=S+aaERy and the operations 0 and on the cosets are defined by(S+a)e(S+b=S+(a+b); (Sta)(S+b)=S+(a b); then R/;e,& is a ring. Proof: Because s; tl is a normal subgroup of r; +, R/S; e is a group Because r; is a commutative group R/S, e is also a commutative group Need prove R/s; is an algebraic system a semigroup, distributive laws

3. Quotient ring Theorem 6.35: Let [R; + ,*] be a ring and let S be an ideal of R. If R/S ={S+a|aR} and the operations  and  on the cosets are defined by (S+a)(S+b)=S+(a + b) ; (S+a)(S+b) =S+(a*b); then [R/S;  ,  ] is a ring. Proof: Because [S;+] is a normal subgroup of [R;+], [R/S;] is a group. Because [R;+] is a commutative group, [R/S;] is also a commutative group. Need prove [R/S;] is an algebraic system, a sumigroup, distributive laws

Definition 32 Under the conditions of Theorem 6.35, R/S;e, 6 is a ring which is called a quotient ring. ◆ Example:LetZ(i)={a+bia,b∈}, E(=2a+2bia, bEZ. E(; + is an ideal of ring Z(i; + Is Quotient ring ZO/EQ:0,6 a field? ◆zero- divisor!

 Definition 32: Under the conditions of Theorem 6.35, [R/S;  , ] is a ring which is called a quotient ring. Example:Let Z(i)={a+bi|a,bZ}, E(i)={2a+2bi|a,bZ}. [E(i);+,*] is an ideal of ring [Z(i);+,*]. Is Quotient ring [Z(i)/E(i);  , ] a field?  zero-divistor!

t EXample: Let r be a commutative ring, andh be an ideal of r. prove that quotient ring r/h is an integral domain For any a,b∈R,ifab∈H, then a∈Hor b∈H Proof: (1)If quotient ring R/H is an integral domain, then a∈Horb∈ H when ab∈ H for any a,b∈R .(2)R is a commutative ring, and h be an ideal of f,Ifa∈Horb∈ H when ab∈H for any a, bER, then quotient ring r/h is an integral domain

Example: Let R be a commutative ring, and H be an ideal of R. Prove that quotient ring R/H is an integral domain  For any a,bR, if abH, then aH or bH.  Proof: (1)If quotient ring R/H is an integral domain, then aH or bH when abH for any a,bR.  (2)R is a commutative ring, and H be an ideal of R. If aH or bH when abH for any a,bR, then quotient ring R/H is an integral domain

e Definition 33: Let p be a ring homomorphism from ring r; + to ring 号,兴 S;+,*]. The kernel of p is the set ker={x∈Rp(x)=0s} A Theorem 6.36: Let (p be a ring s homomorphism from ring [r;+, to ring S;+’,*],Then .(D(P(R); + is a subring of [s; +,*1 .(2)lker; +, * is an ideal of r; + I

 Definition 33: Let  be a ring homomorphism from ring [R;+,*] to ring [S;+’,*’]. The kernel of  is the set ker={xR|(x)=0S }.  Theorem 6.36: Let  be a ring homomorphism from ring [R;+,*] to ring [S;+’,*’]. Then  (1)[(R);+’,*’] is a subring of [S;+’,*’]  (2)[ker;+,*] is an ideal of [R;+,*]