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;+,*]