‫ﺟﺒﺮ اﻟﺤﻠﻘﺎت‬
‫اﻟﻤﺤﺎﺿﺮة اﻟﺨﺎﻣﺴﺔ‬
‫ اﻟﻤﺜﺎﻟﯿﺔ ﻓﻲ‬. ( ideal)‫ﻓﯿﻤﺎ ﯾﻠﻲ ﺳﻨﺘﻄﺮق إﻟﻰ ﻧﻮع ﻣﮭﻢ ﻣﻦ اﻟﺤﻠﻘﺎت اﻟﺠﺰﺋﯿﺔ ﯾﺴﻤﻰ اﻟﻤﺜﺎﻟﯿﺔ‬
‫ ﺣﯿﺚ ﺳﯿﺆدي‬،‫( ﻓﻲ اﻟﺰﻣﺮة‬normal subgroup) ‫اﻟﺤﻠﻘﺔ ﻟﮭﺎ ﻧﻔﺲ ﻣﻮﻗﻊ اﻟﺰﻣﺮة اﻟﺠﺰﺋﯿﺔ اﻟﺴﻮﯾﺔ‬
.(quotient ring) ‫وﺟﻮدھﺎ إﻟﻰ ﺑﻨﺎء ﺣﻠﻘﺔ اﻟﻘﺴﻤﺔ‬
Definition 1.9. A subring I of the ring R is said to be a two-sided ideal of
R if and only if r Î R and aÎ I imply that both ra Î I and arÎ I.
:‫ ﯾﻤﻜﻦ إﻋﺎدة ﺻﯿﺎﻏﺔ اﻟﺘﻌﺮﯾﻒ اﻷﺧﯿﺮ ﻛﻤﺎ ﯾﻠﻲ‬1.3 ‫ﺑﺎﻻﺳﺘﻨﺎد إﻟﻰ ﻣﺒﺮھﻨﺔ‬
Definition 1.10. Let I be a nonempty subset of a ring R. Then I is a twosided ideal of R if and only if
1. a, b Î I imply a - b Î I and 2. rÎ R and aÎ I imply that both ra, ar Î I.
‫( ﻓﺎﻟﻤﺜﺎﻟﯿﺔ ﺗﻜﻮن ﻣﺜﺎﻟﯿﺔ‬ra Î I) ‫ ﻓﻲ ﺣﺎﻟﺔ اﺧﺘﺼﺎر اﻟﺸﺮط اﻟﺜﺎﻧﻲ ﻓﻲ ﺗﻌﺮﯾﻒ اﻟﻤﺜﺎﻟﯿﺔ إﻟﻰ‬: ‫ﻣﻼﺣﻈﺔ‬
‫ ﻟﻜﻦ ﻟﻨﺘﻔﻖ ﻓﯿﻤﺎ ﯾﻠﻲ ﻣﻦ‬. (right ideal) ‫( وﺑﻨﻔﺲ اﻟﻄﺮﯾﻖ ﻧﻌﺮف ﻣﺜﺎﻟﯿﺔ ﯾﻤﯿﻦ‬left ideal)‫ﯾﺴﺎر‬
.(two-sided ideal) ‫(ﻓﺈﻧﻨﺎ ﻧﻘﺼﺪ ﻣﺜﺎﻟﯿﺔ ﻣﻦ اﻟﺠﮭﺘﯿﻦ‬ideal) ‫اﻟﺒﺤﺚ ﻋﻨﺪﻣﺎ ﻧﺬﻛﺮ ﻣﺜﺎﻟﯿﺔ‬
.‫ﻗﺒﻞ ان ﻧﺴﺘﻤﺮ ﺑﺎﻟﺒﺤﺚ ﻟﻨﻌﺰز اﻟﺘﻌﺮﯾﻒ ﺑﺒﻌﺾ اﻷﻣﺜﻠﺔ‬
Example 1.13
With respect to the usual matrix operations, the set
ì éa b ù
ü
T2 ( Z) = íê
| a , b, c Î Zý
ú
î ë0 c û
þ
forms a noncommutative ring with identity ( in fact, a subring of the ring
M 2 (Z) of all 2 ´ 2 matrices with integer entries). We propose to show that
in this ring the subset
ì éa b ù
ü
I = íê
| a , b Î Zý
ú
î ë0 0 û
þ
is an ideal. I is a right ideal of the ring M 2 (Z), but not ideal.
(‫) اﻟﺘﻮﺿﯿﺢ ﻓﻲ اﻟﺼﻒ‬
Example 1.14
For each integer a Î Z, let (a) represent the set consisting of all integral
multiples of a; that is (a) ={ na| nÎ Z}. (a) is an ideal of Z.
Example 1.15
Let R be a ring, X a nonempty set and x a fixed element of X. The set
I x = {fÎ map(X, R) | f(x)= 0}
is an ideal of the ring map(X, R) .
More generally, if S is any nonempty subset of X, then
I = { f Î map(X, R) | f(x)= 0 for all xÎ S}
is also an ideal of the ring map(X, R), in fact I= IxÎS I x .
Theorem 1.10 If I is a proper (right, left, two-sided) ideal of a ring R
with identity, then no element of I possesses a multiplicative inverse.
Proof: (in the class).
:‫ﺳﻨﻼﺣﻆ أن اﻟﺒﺮھﺎن ﯾﺘﻀﻤﻦ اﻟﻨﺘﯿﺠﺔ اﻟﺘﺎﻟﯿﺔ‬
Corollary. In a ring with identity, no proper (right, left, two sided) ideal
contains the identity element.
.‫واﻵن ﺳﻨﺪرس إﻣﻜﺎﻧﯿﺔ اﻟﺤﺼﻮل ﻋﻠﻰ ﻣﺜﺎﻟﯿﺎت ﺟﺪﯾﺪة ﻣﻦ ﻣﺜﺎﻟﯿﺎت ﻣﻌﻠﻮﻣﺔ‬
Theorem 1.11 Let { I i } be an arbitrary collection of (right, left, twosided) ideals of the ring R, where i ranges over some index set. Then
I I i is also a (right, left, two-sided) ideal of R.
i
Proof: (in the class).
.‫ﻓﯿﻤﺎ ﯾﻠﻲ ﺗﻌﺮﯾﻒ اﻟﻤﺜﺎﻟﯿﺔ اﻟﻤﺘﻮﻟﺪة ﺑﻮاﺳﻄﺔ ﻣﺠﻤﻮﻋﺔ ﺟﺰﺋﯿﺔ ﻏﯿﺮ ﺧﺎﻟﯿﺔ ﻣﻦ ﺣﻠﻘﺔ ﻣﻌﻠﻮﻣﺔ‬
Definition 1.11 Let R be a ring, S a nonempty subset of R. By the
symbol ( S ) we shall mean the set
( S )= Ç { I| S Í I ; I is an ideal of R}
(S) is an ideal ( by Theorem 1.11), in fact it is the smallest ideal of R
which contain S and it is called the ideal generated by the set S.
If S consist of a finite number of elements, say a 1 , a 2 , ..., a n , then the
ideal that they generated is denoted by ( a 1 , a 2 , ..., a n ). Such an ideal is said
to be finitely generated with the given elements a i as its generators. An
ideal (a) generated by just one element is termed a principal ideal.
a ‫ ( اﻟﻤﺘﻮﻟﺪة ﺑﻌﻨﺼﺮ واﺣﺪ‬principal ideal ) ‫ﻓﯿﻤﺎ ﯾﻠﻲ ﻧﻮﺻﻒ ﻋﻨﺎﺻﺮ اﻟﻤﺜﺎﻟﯿﺎت اﻟﺮﺋﯿﺴﯿﺔ‬
‫ ﯾﺴﺎر وﻣﻦ اﻟﺠﮭﺘﯿﻦ‬، ‫ﯾﻤﯿﻦ‬
( a) r = { ar + na | r Î R; nÎ Z}
( a) l = { ra + na | r Î R; nÎ Z}
(a ) = { na + ra + as + å ri as i | r, s, ri , s i Î R; n Î Z }
finite
If R is a ring with identity, then
( a) r = aR= { ar | r Î R}; ( a) l = Ra={ ra | r Î R} and
(a ) = {
å r as
i
i
| r, s, ri , s i Î R }
finite
: ‫اﻟﻤﺜﺎﻟﯿﺎت اﻟﺮﺋﯿﺴﯿﺔ ﺳﺘﻘﻮدﻧﺎ إﻟﻰ ﻧﻮع ﻣﮭﻢ ﻣﻦ اﻟﺤﻠﻘﺎت ﺗﺴﻤﻰ ﺣﻠﻘﺔ اﻟﻤﺜﺎﻟﯿﺎت اﻟﺮﺋﯿﺴﯿﺔ‬
Definition 1.12 A ring R is said to be a principal ideal ring if every ideal
I of R is of the form I = (a) for some a Î R.
.‫اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ ﺗﻘﺪم ﻟﻨﺎ ﻣﺜﺎﻻ ﻋﻦ ھﻜﺬا ﺣﻠﻘﺔ‬
Theorem 1.12 The ring Z of integers is a principal ideal ring; in fact, if I
is an ideal of Z, then I = (n) for some nonnegative integer n.
Proof: (in the class)
‫اﻟﻤﺤﺎﺿﺮة اﻟﺴﺎدﺳﺔ‬
‫ﻓﻲ اﻟﻔﻘﺮة اﻟﺴﺎﺑﻘﺔ ﻻﺣﻈﻨﺎ أن ﺗﻘﺎطﻊ أي ﻋﺪد)ﻣﻨﺘﮭﻲ أو ﻏﯿﺮ ﻣﻨﺘﮭﻲ( ﻣﻦ اﻟﻤﺜﺎﻟﯿﺎت ﯾﻜﻮن ﻣﺜﺎﻟﯿﺔ‬
‫ ﻣﻦ اﻟﻄﺒﯿﻌﻲ أن ﻧﺴﺄل ﻣﺎذا ﯾﺤﺼﻞ ﻓﻲ ﺣﺎﻟﺔ اﺗﺤﺎد اﻟﻤﺜﺎﻟﯿﺎت وﺳﯿﻜﻮن اﻟﺠﻮاب ﻧﻔﯿﺎ ﺑﺸﻜﻞ‬. ‫أﯾﻀﺎ‬
‫ﻋﺎم ) ﻧﻌﻄﻲ ﻣﺜﺎل ﻓﻲ اﻟﺼﻒ( وﻟﻜﻦ ﯾﻤﻜﻦ ان ﻧﺤﺼﻞ ﻋﻠﻰ ﻣﺜﺎﻟﯿﺔ ﻣﻦ اﺗﺤﺎد ﻋﺪد ﻣﺘﻨﺎھﻲ ﻣﻦ‬
‫اﻟﻤﺜﺎﻟﯿﺎت وھﻲ اﻟﻤﺜﺎﻟﯿﺔ اﻟﻤﺘﻮﻟﺪة ﺑﻮاﺳﻄﺔ اﻻﺗﺤﺎد وﻓﻲ ھﺬه اﻟﺤﺎﻟﺔ ﺳﻨﺴﻤﻲ اﻟﻤﺜﺎﻟﯿﺔ اﻟﺠﺪﯾﺪة ﻣﺠﻤﻮع‬
. ‫اﻟﻤﺜﺎﻟﯿﺎت‬
Definition 1.13 Given a finite number of ideals I 1 , I 2 , . . . , I n of the ring
R , then their sum is defined :
I 1 + I 2 + . . . + I n = { a 1 + a 2 + . . . +a n | a i Î I i } it is an ideal, in fact it
is the ideal generated by the union I 1 È I 2 È . . . È I n . In the special case
of two ideals I and J or definition reduces to I + J = { a + b| a Î ; bÎ J}.
More generally, let {I i } be an arbitrary indexed collection of ideals of
R. The sum of this collection may be denoted by å I i and is the ideal of R
whose members are all possible finite sums of elements from the various
ideals I i : å I i = { å a i | a i Î I i }.
finite
‫ ﺗﻘﺎطﻊ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻤﺜﺎﻟﯿﺎت ھﻮ اﻛﺒﺮ ﻣﺜﺎﻟﯿﺔ ﻣﺤﺘﻮاة ﻓﻲ ﻛﻞ ﻣﺜﺎﻟﯿﺔ ﻣﻦ ﻣﺜﺎﻟﯿﺎت ﺗﻠﻚ‬:‫ﻣﻼﺣﻈﺔ‬
‫ ﺑﺎﻟﻤﻘﺎﺑﻞ ﻓﺎن ﻣﺠﻤﻮع اﻟﻤﺜﺎﻟﯿﺎت ﻓﻲ ﻣﺠﻤﻮﻋﺔ ﻣﺎ ھﻮ اﺻﻐﺮ ﻣﺜﺎﻟﯿﺔ ﺗﺤﺘﻮي ﺟﻤﯿﻊ‬،‫اﻟﻤﺠﻤﻮﻋﺔ‬
.‫اﻟﻤﺜﺎﻟﯿﺎت ﻓﻲ ﺗﻠﻚ اﻟﻤﺠﻤﻮﻋﺔ‬
.‫واﻵن ﻧﺘﻄﺮق إﻟﻰ ﻧﻮع آﺧﺮ ﻣﻦ إﻧﺘﺎج اﻟﻤﺜﺎﻟﯿﺔ ﻣﻦ ﻣﺜﺎﻟﯿﺎت ﻣﻌﻠﻮﻣﺔ أﻻ وھﻮ ﺿﺮب اﻟﻤﺜﺎﻟﯿﺎت‬
Definition 1.14 Given two ideals I and J of a ring R, we define
IJ={
åa b
i
i
| a i Î I; b i Î J }.
finite
with this definition, IJ indeed becomes an ideal of R. ( details in class)
In fact, I J is just the ideal generated by the set of all products ab,
a Î I; bÎ J.
‫ وﻋﻠﻰ‬I 1 ,I 2 ,...,I n ‫ﻻ ﺗﻮﺟﺪ ﺻﻌﻮﺑﺔ ﻓﻲ ﺗﻌﻤﯿﻢ ﺿﺮب اﻟﻤﺜﺎﻟﯿﺎت اﻟﻰ ﻋﺪد ﻣﺘﻨﺎه ﻣﻦ اﻟﻤﺜﺎﻟﯿﺎت‬
‫ ﻣﻦ اﻟﻤﺮات ﻓﻨﺤﺼﻞ ﻋﻠﻰ‬n ‫ ﻓﻲ ﻧﻔﺴﮭﺎ‬I ‫اﻟﺨﺼﻮص ﻓﻲ ﺿﺮب ﻣﺜﺎﻟﯿﺔ‬
n
I = { å (a i1a i 2 ... a in ) | a ik Î I }.
finite
‫ﻋﻤﻠﯿﺔ اﻟﻀﺮب ﺑﯿﻦ اﻟﻤﺜﺎﻟﯿﺎت ﺳﺘﻘﻮدﻧﺎ اﻟﻰ ﺗﻌﺮﯾﻒ أﻧﻮاع ﺧﺎﺻﺔ ﻣﻦ اﻟﻤﺜﺎﻟﯿﺎت ﻟﮭﺎ أھﻤﯿﺘﮭﺎ‬
Definition 1.15 An ideal I of a ring R is said to be a nil ideal if each
element a Î I is nilpotent; that is, if there exists a positive integer n for
which a n =0, where n depends on the particular element a.
The ideal I will be termed nilpotent provided that I n ={0} for some
positive integer n.
Note: The condition I n ={0} is equivalent to requiring that for every
choice of n elements a 1 , a 2 , …, a n Î I (distinct or not), the product
a 1 a 2 …a n =0. In particular, a n =0 for all a in I, whence every nilpotent
ideal is automatically nil ideal.
Example 1.16
I = {0, 2, 4} is a nilpotent ideal of the ring Z 8 of integers modulo 8; here,
it will be found that I 3 ={0}. More generally, the principal ideal
(p) = pZ p , for any prime p and n>1.
n
.‫ﺑﺤﺜﻨﺎ اﻟﻘﺎدم ﺳﯿﺘﻀﻤﻦ ﺗﺄﺛﯿﺮ اﻟﺘﺸﺎﻛﻞ ﻋﻠﻰ اﻟﻤﺜﺎﻟﯿﺔ‬
Theorem 1.13 Let f be a homomorphism from the ring R onto the ring
R ¢ . Then
a) for each ideal I of R, f(I) is an ideal of R ¢ ; and
b) for each ideal I¢ of R ¢ , f -1 ( I¢ ) is an ideal of R.
Proof: in the class.
Definition 1.16 Let f be a homomorphism from the ring R into the ring
R ¢ , then the kernel of f is the set ker f = {a Î R| f(a) = 0}.
‫ وﻛﻤﺎ ﻓﻲ ﺣﺎﻟﺔ‬. ‫ ھﻮ ﻧﻔﺲ اﻟﺘﻌﺮﯾﻒ ﻓﻲ ﺣﺎﻟﺔ اﻟﺰﻣﺮ‬ker f ‫اذا اھﻤﻠﻨﺎ ﻋﻤﻠﯿﺔ اﻟﻀﺮب ﻓﺎن ﺗﻌﺮﯾﻒ‬
.ker f ={0} ‫( اذا وﻓﻘﻂ اذا ﻛﺎن‬1-1) ‫ ﯾﻜﻮن ﻣﺘﺒﺎﯾﻦ‬f ‫اﻟﺰﻣﺮ ﻓﺎن‬
.‫ﻻﺣﻆ اﯾﻀﺎ اﻟﻨﺘﺎﺋﺞ اﻟﺘﺎﻟﯿﺔ اﻟﻤﺸﺎﺑﮫ ﻟﻨﺘﺎﺋﺞ ﻓﻲ ﻧﻈﺮﯾﺔ اﻟﺰﻣﺮ‬
Theorem 1.14 The kernel ker f of a homomorphism f from a ring R into
a ring R ¢ is an ideal of R.
‫اﻟﻤﺤﺎﺿﺮة اﻟﺴﺎﺑﻌﺔ‬
Theorem 1.15( Correspondence Theorem) Let f be a homomorphism
from the ring R onto the ring R ¢ . Then there is a one-to-one correspondence between those ideals I of R such that ker f Í I and the set of ideals I¢
of R ¢ ; specifically, I¢ is given by I¢ =f(I).
Example 1.17
Let R be any ring with identity and let f: Z→R be a mapping defined by,
f(n)=n1 (the n-fold sum of 1). Then
1. f is a homomorphism;
2. ker f = {nÎ Z| n1 = 0}= {p}, for some nonnegative integer p;
3. In particular any ring with identity that is of characteristic zero will
contain a subring isomorphic to the integers (Z @ Z1).
.‫ أﻻ وھﻮ إﻧﺘﺎج ﺣﻠﻘﺔ اﻟﻘﺴﻤﺔ‬، ‫ﻓﯿﻤﺎ ﯾﻠﻲ ﺳﻨﺒﺤﺚ اﻟﺪور اﻷﺳﺎﺳﻲ ﻟﻠﻤﺜﺎﻟﯿﺔ‬
Definition 1.17 Given an ideal I of the ring R we have
1) a+I = {a+i| iÎ I}; 2) R/I = { a+I| a Î R}; 3) a+I=b+I ↔ a-bÎ I;
4) (a+I) + (a+I) = (a+b) + I; 5) (a+I)(a+I)= (ab)+I.
‫ﺎ‬l‫ﺔ ﻛﻤ‬l‫ﺮة اﺑﺪاﻟﯿ‬l‫ﻞ زﻣ‬l‫( ﯾﻤﺜ‬R/I, +) ‫ﺢ ان‬l‫ﻣﻦ ﺧﻼل دراﺳﺘﻨﺎ ﻟﻠﺰﻣﺮ واﻟﺘﻌﺮﯾﻒ اﻋﻼه ﻣﻦ اﻟﻮاﺿ‬
‫ﺔ‬ll‫ﻊ ﻋﻤﻠﯿ‬l‫ﯿﺔ ﺗﻮزﯾ‬l‫ﻦ ﺧﺎﺻ‬l‫ﺮة وﻣ‬l‫ﺒﮫ زﻣ‬l‫ﻞ ﺷ‬l‫( ﯾﻤﺜ‬R/I, ∙) ‫ﻮن‬l‫ﻦ ﻛ‬l‫ﻖ ﻣ‬l‫ﺼﻌﺐ اﻟﺘﺤﻘ‬l‫ﻦ اﻟ‬l‫ﯿﺲ ﻣ‬l‫ﮫ ﻟ‬l‫اﻧ‬
‫اﻟﻀﺮب ﻋﻠﻰ ﻋﻤﻠﯿﺔ اﻟﺠﻤﻊ وﺑﺬﻟﻚ ﻧﺤﺼﻞ ﻋﻠﻰ اﻟﺤﻘﯿﻘﺔ اﻟﺘﺎﻟﯿﺔ‬
Theorem 1.16 If I is an ideal of the ring R, then R/I is also a ring,
known as the quotient ring of R by I.
Example 1.18
Consider Z 8 the ring of integers modulo 8, and the ideal I ={0,4}. It is
easily seen that the quotient ring Z 8 /I has four elements, namely, the
cosets 0+I = {0,4}, 1+I = {1,5}, 2+I = {2,6} and 3+I = {3,7}.
‫ﺪ‬l‫ﻦ ﻧﺠ‬l‫ﺎل وﻟ‬l‫ﺬا اﻟﻤﺜ‬l‫ﻲ ھ‬l‫ﺴﻤﺔ ﻓ‬l‫ﺔ اﻟﻘ‬l‫ﻀﺮب ﻟﺤﻠﻘ‬l‫ﻊ واﻟ‬l‫ﻲ اﻟﺠﻤ‬l‫ﺪاول ﻋﻤﻠﯿﺘ‬l‫ﻓﻲ اﻟﺼﻒ ﺳﻨﺘﺤﻘﻖ ﻣﻦ ﺟ‬
. Z 4 ‫ﺻﻌﻮﺑﺔ ﻟﻨﻼﺣﻆ اﻧﮭﺎ ﻣﻜﺎﻓﺌﺔ ﻟﻠﺤﻠﻘﺔ‬
Example 1.19
In the ring Z of integers, consider the principal ideal (n), where n is a
nonnegative integer. The cosets of (n) in Z take the form
a + (n) ={a + kn | k Î Z}, so, Z/(n) is just the ring Z n of integers modulo n.
(‫) ﻣﺎ ھﻲ؟‬.‫اﻟﻤﺒﺮھﻨﺔ اﻟﻘﺎدﻣﺔ ﺳﺒﻖ وان ﻣﺮ ﻋﻠﯿﻨﺎ ﻣﺒﺮھﻨﺔ ﻧﻈﯿﺮة ﻋﻨﺪﻣﺎ درﺳﻨﺎ اﻟﺰﻣﺮة‬
Theorem 1.20 Any ideal of a ring R is the kernel of some
homomorphism from R onto certain ring R ¢ .
Proof: In the class.
‫ﺘﻨﺎ‬l‫ﺪ دراﺳ‬l‫ﺎ ﻋﻨ‬l‫واﻻن ﺟﺎء وﻗﺖ اﻟﻤﺒﺮھﻨﺎت اﻻﺳﺎﺳﯿﺔ ﻟﻠﺘﺸﺎﻛﻞ ﻓﻲ اﻟﺤﻠﻘﺎت واﻟﺘﻲ ﻣﺮ ﻋﻠﯿﻨﺎ ﻧﻈﺎﺋﺮ ﻟﮭ‬
.‫ وﻧﺒﺪأ ﺑﺎﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ‬.‫ﻟﻠﺰﻣﺮ‬
Theorem 1.21 (Fundamental Homomorphism Theorem) If f is a
homomorphism from the ring R onto the ring R ¢ , then R/ker f @ R ¢ .
Proof: In the class.
‫ﻣﻦ اﺟﻞ ﻟﻤﺲ ﻓﻜﺮة اﻟﻤﺒﺮھﻨﺔ اﻟﺴﺎﺑﻘﺔ ﻧﺪرس اﻟﻤﺜﺎل اﻟﺘﺎﻟﻲ‬
Example 1.20
Let f: Z 4 → Z 2 be define by f(0)=f(2)=0; f(1)=f(3)=1. ( other details in
class)
‫ﺔ‬ll‫ﻰ ﺣﻠﻘ‬ll‫ﺔ اﻟ‬ll‫ﻦ أي ﺣﻠﻘ‬ll‫ﺸﺎﻛﻼت ﻣ‬ll‫ﻮل اﻟﺘ‬ll‫ ﺣ‬1.21 ‫ﺔ‬ll‫ﺎت ﻣﺒﺮھﻨ‬ll‫ﺪ ﺗﻄﺒﯿﻘ‬ll‫ﺢ اﺣ‬ll‫ﺔ ﺗﻮﺿ‬ll‫ﺔ اﻟﺘﺎﻟﯿ‬ll‫اﻟﻤﺒﺮھﻨ‬
‫ وﻟﻜﻦ ﻗﺒﻠﮭﺎ ﻧﺤﺘﺎج اﻟﻰ اﻟﺘﻤﮭﯿﺪ)ﻣﺄﺧﻮذة( اﻟﺘﺎﻟﻲ‬،‫اﻻﻋﺪاد اﻟﺼﺤﯿﺤﺔ‬
Lemma. The only nontrivial homomorphism from the ring Z of integers
into itself is the identity map i Z .
Proof: In the class.
Corollary. There is at most one homomorphism under which an
arbitrary ring R is isomorphic to the ring Z.
Proof: In the class.
Theorem 1.22 Any homomorphism from an arbitrary ring R onto the
ring the ring Z of integers is uniquely determined by its kernel.
Proof: In the class.
(‫ﺔ‬l‫ﺮة واﻟﺤﻠﻘ‬l‫ ھﺬه اﻟﻤﺒﺮھﻨﺔ ﻟﯿﺲ ﻟﮭﺎ ﻧﻈﯿﺮ ﻓﻲ اﻟﺰﻣﺮ) ﻣﻤﺎ ﯾﺆﺷﺮ اﺣﺪ اﻻﺧﺘﻼﻓﺎت ﺑﯿﻦ اﻟﺰﻣ‬:‫ﻣﻼﺣﻈﺔ‬
.ker i Z =ker - i Z ={0} ‫ وﻟﻜﻦ‬i Z ¹ - i Z ‫ﺣﯿﺚ ان‬
Problems 2.
1. If I is a right and J a left ideal of the ring R such that I Ç J = {0}, prove
that ab = 0 for all a Î I, b Î J.
2. Given an ideal I of the ring R, define the set C(I) by
C(I) = { r Î R | ra – ar Î I for all a Î R}.
Verify that C(I) forms a subring of R.
3. a) Show by example that I and J are both ideals of the ring R, then I È J
need not be an ideal .
b) If { I i } (i =1,2,… ) is a collection of ideals of the ring R such that
I1 Í I 2 Í ... Í I n Í ... , prove that È I i is also an ideal of R.
4. Let R be an arbitrary ring and R ´ Z be the extension ring constructed
in Theorem 1.9. Establish that
a) R ´ {0} is an ideal of R ´ Z;
b) Z @ {0} ´ Z; c) if a is an idempotent element of R, then the pair (-a,1)
is idempotent in, R ´ Z, while (a, 0) is a zero divisor.
5. Let I be an ideal of R, a commutative ring with identity. For an element
a Î R the ideal generated by the set I È {a} is denoted by (I, a). Assuming
that a Ï I, show that (I, a) = {I + ra | i Î I, r Î R}.
6. Establish each of the assertions below:
a) M 2 (Z e ) is an ideal of the matrix ring M 2 (Z).
b) The set
é0 a ù
I = { ê ú | aÎ R # }
ë0 0 û
is an ideal of the ring T 2 ( R # ) of 2 ´ 2 triangular matrices.
c) The matrix ring M 2 ( R # ) has no nontrivial ideals.
7. Show that Z is a subring of the ring Q of rational numbers but not an
ideal of Q.
8. Prove that every subring of Z forms an ideal. Do the same for the ring
Zn .
9. Consider the set R = { a + bi | a, b Î Z}, where i 2 = -1. Verify that R
forms a ring with respect to ordinary addition and multiplication and that
I = { a +2bi | a, b Î Z} is an ideal of this ring.
10. a) Determine whether Z ´ Z e is an ideal ring Z ´ Z.
b) In the ring P(X) of all subsets of a set X, show that the collection of
all finite subsets of X forms an ideal.