‫ﺟﺒﺮ اﻟﺤﻠﻘﺎت‬
‫اﻟﻤﺤﺎﺿﺮة اﻷوﻟﻰ‬
‫ﻣﻦ اﺟﻞ ﻓﺎﺋﺪة اﻟﻄﺎﻟﺐ ﺳﺘﻜﻮن ﻟﻐﺔ ﻣﺤﺎﺿﺮات ﺟﺒﺮ اﻟﺤﻠﻘﺎت ھﻲ اﻟﻠﻐﺔ اﻻﻧﺠﻠﯿﺰﯾﺔ ﻣﻊ اﺳﺘﺨﺪام‬
‫اﻟﻠﻐﺔ اﻟﻌﺮﺑﯿﺔ ﻟﻠﺘﻮﺿﯿﺤﺎت واﻟﺘﻌﻠﯿﻘﺎت‪.‬‬
‫درس اﻟﻄﺎﻟﺐ ﻓﻲ اﻟﻤﺮﺣﻠﺔ اﻟﺜﺎﻧﯿ ﺔ ﺟﺒ ﺮ اﻟﺰﻣ ﺮ وﺗﻌ ﺮف ﻋﻠ ﻰ اﻟﺰﻣ ﺮة ﻋﻠ ﻰ أﻧﮭ ﺎ ﻧﻈ ﺎم ﺟﺒ ﺮي‬
‫ﯾﺘﻜﻮن ﻣﻦ ﻣﺠﻤﻮﻋﺔ ﺧﺎﻟﯿﺔ ﻣﻊ ﻋﻤﻠﯿﺔ ﺛﻨﺎﺋﯿﺔ ﺗﺤﻘﻖ ﺷﺮوط ﻣﻌﯿﻨﺔ ‪ ،‬وﻻﺣﻆ اﻟﻄﺎﻟﺐ أن ﻣﻔﮭﻮم اﻟﺰﻣ ﺮة‬
‫ﺟﺎء ﻣﻦ ﻧﻈﺎم اﻷﻋﺪاد ‪ .‬ﻓﻲ ھﺬه اﻟﻤﺮﺣﻠﺔ ﺳﻨﺪرس وﻧﺘﻌﺮف ﻋﻠﻰ ﻧﻈﻢ ﺟﺒﺮﯾﺔ أﺧﺮى اﻗﺮب إﻟﻰ ﻧﻈﺎم‬
‫اﻷﻋﺪاد ﻣﺜﻞ اﻟﺤﻠﻘﺔ )‪ (ring‬واﻟﺤﻘﻞ)‪ (field‬واﻟﺘﻲ ﺗﺘﻜﻮن ﻣﻦ ﻣﺠﻤﻮﻋﺔ ﺧﺎﻟﯿ ﺔ ﻣ ﻊ ﻋﻤﻠﯿﺘ ﯿﻦ ﺛﻨﺎﺋﯿ ﺔ‬
‫ﺗﺤﻘﻖ ﺷﺮوط ﻣﻌﯿﻨﺔ‪ .‬وﺳﻨﺒﺪأ ﺑﺎﻟﺤﻠﻘﺔ واﻟﺘﻲ ھﻲ اﻋﻢ ﻣﻦ اﻟﺤﻘﻞ )ﻛﻞ ﺣﻘﻞ ھﻮ ﺣﻠﻘﺔ وﻟﯿﺲ اﻟﻌﻜﺲ(‪.‬‬
‫ﻧﻘﻮل ﺑﺒﺴﺎطﺔ أن اﻟﺤﻠﻘﺔ ھﻲ ﺗﺮﻛﯿﺐ ﻣﻦ زﻣﺮة اﺑﺪاﻟﯿﺔ )ﻟﻠﻌﻤﻠﯿﺔ اﻻوﻟﻰ( وﺷﺒﮫ زﻣﺮة )ﻟﻠﻌﻤﻠﯿﺔ اﻟﺜﺎﻧﯿﺔ(‬
‫ﻣ ﻊ ﻋﻼﻗ ﺔ ﺗ ﺮﺑﻂ ﺑ ﯿﻦ اﻟﻌﻤﻠﯿﺘ ﯿﻦ‪ ،‬ﺣﯿ ﺚ أن اﻟﻌﻤﻠﯿ ﺔ اﻟﺜﺎﻧﯿ ﺔ ﺗﺘ ﻮزع ﻋﻠ ﻰ اﻻوﻟ ﻰ‪ .‬أدﻧ ﺎه ﺳ ﻨﻌﻄﻲ‬
‫اﻟﺘﻌﺮﯾﻔﺎت اﻟﺪﻗﯿﻘﺔ ﻟﮭﺬه اﻟﻤﻔﺎھﯿﻢ‪.‬‬
‫‪Definition 1.1 Given an algebraic system (S, ▫ , ◦), the operation ◦ is said‬‬
‫‪to be left distributive over ▫ if‬‬
‫)‪a ◦ (b▫c) (a◦b) ▫ (a◦c‬‬
‫‪and right distributive if‬‬
‫)‪(b▫c) ◦ a (b◦a) ▫ (c ◦a‬‬
‫‪for all elements a, b, c Î S. The operation ◦ is distributive over ▫ if both‬‬
‫‪of these conditions hold.‬‬
‫إذا ﻛﺎﻧﺖ اﻟﻌﻤﻠﯿﺔ ◦ اﺑﺪاﻟﯿﺔ ﻓﯿﻜﻔﻲ اﺣﺪ اﻟﺸﺮطﯿﻦ أﻋﻼه ﻟﺘﺤﻘﻖ ﺧﺎﺻﯿﺔ اﻟﺘﻮزﯾﻊ )اﻟﻤﻄﻠﻘﺔ(‪.‬‬
‫‪Definition 1.2 A ring is an ordered triple ( R, +, ∙ ) consisting of a‬‬
‫‪nonempty set R and two binary operations + and ∙ defined on R such that‬‬
‫‪1. (R, +) is a commutative group,‬‬
‫‪2. (R, ∙) is a semigroup, and‬‬
‫‪3. the operation ∙ is distributive over the operation + .‬‬
‫ﯾﻨﺒﻐﻲ أن ﯾﻔﮭﻢ اﻟﻄﺎﻟﺐ أن اﻟﻌﻤﻠﯿﺘﺎن ‪ +‬و ∙ ﻓﻲ اﻟﺘﻌﺮﯾﻒ أﻋﻼه ﻻ ﯾﻘﺼﺪ ﺑﮭﺎ ﻋﻤﻠﯿﺘﺎ اﻟﺠﻤﻊ واﻟﻀﺮب‬
‫اﻻﻋﺘﯿﺎدﯾﺔ وإﻧﻤﺎ ﻋﻤﻠﯿﺘﺎن ﻣﺠﺮدة ‪ .‬وﻣﻊ ذﻟﻚ ﻓﺎن ﻣﻦ اﻟﻤﻨﺎﺳﺐ أن ﻧﺴﺘﺨﺪم اﻟﺮﻣﺰ ‪ 0‬ﻟﻠﻌﻨﺼﺮ‬
‫اﻟﺼﻔﺮي ﻟﻠﻌﻤﻠﯿﺔ ‪ +‬وﻛﺬﻟﻚ اﻟﺮﻣﺰ ‪ –a‬ﻧﻈﯿﺮ اﻟﻌﻨﺼﺮ ‪ a‬ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻌﻤﻠﯿﺔ ‪ .+‬ﺳﻨﻌﯿﺪ ﺻﯿﺎﻏﺔ ﺗﻌﺮﯾﻒ‬
‫اﻟﺤﻠﻘﺔ أﻋﻼه ﻛﻤﺎ ﯾﻠﻲ‪.‬‬
‫‪A ring ( R, +, ∙ ) consists of a nonempty set R together with two binary‬‬
‫‪operations + and ∙ of addition and multiplication on R for which the‬‬
‫‪following conditions are satisfied :‬‬
‫‪1. a + b = b+ a,‬‬
‫‪2. (a +b) + c = a +(b +c),‬‬
‫‪3. there exists an element 0 in R such that a +0=a for every a Î R,‬‬
‫‪4. for each a Î R there exists an element –aÎ R such that a +(-a) =0,‬‬
‫‪5. (a∙ b) ∙c = a ∙ (b∙ c), and‬‬
6. a∙ ( b+ c) = a∙ b + a∙ c and (b +c) ∙ a = b∙ a + c ∙ a.
‫ ﻋﻨﺪﻣﺎ ﻧﻀﻊ ﺷﺮوط‬.‫ ﻣﻨﺘﮭﯿﺔ‬R ‫( إذا ﻛﺎﻧﺖ اﻟﻤﺠﻤﻮﻋﺔ‬finite) ‫ ( ﻣﻨﺘﮭﯿﺔ‬R, +, ∙ ) ‫ﺗﺴﻤﻰ اﻟﺤﻠﻘﺔ‬
‫إﺿﺎﻓﯿﺔ ﻋﻠﻰ ﻋﻤﻠﯿﺔ اﻟﻀﺮب ) اﻟﻌﻤﻠﯿﺔ ∙( ﻧﺤﺼﻞ ﻋﻠﻰ أﻧﻮاع ﺧﺎﺻﺔ ﻣﻦ اﻟﺤﻠﻘﺎت ﻛﻤﺎ ﻓﻲ اﻟﺘﻌﺮﯾﻒ‬
.‫اﻟﺘﺎﻟﻲ‬
Definition 1.3
1. A commutative ring ( R, +, ∙ ) in which multiplication is a
commutative operation, a∙ b = b∙ a for all a, b Î R. (In case a b= b a
for a particular pair a, b we express this fact by saying a and b
commute)
2. A ring with identity is a ring ( R, +, ∙ ) in which there exists an
identity element for the operation of multiplication normally
represented by the symbol 1, so that a∙1= 1∙ a= a for all aÎ R.
‫( إذا ﻛﺎن ﻟﮫ‬invertible or unit)‫ ذو ﻣﻌﻜﻮس‬a ‫ﻓﻲ اﻟﺤﻠﻘﺔ ذات اﻟﻤﺤﺎﯾﺪ ﻧﻘﻮل أن اﻟﻌﻨﺼﺮ‬
a ‫ ﺑﺸﻜﻞ ﻋﺎم اﻟﻨﻈﯿﺮ أﻟﻀﺮﺑﻲ ﻟﻠﻌﻨﺼﺮ‬. ∙ ‫ﻧﻈﯿﺮ )ﻣﻦ اﻟﻄﺮﻓﯿﻦ( ﺑﺎﻟﻨﺴﺒﺔ إﻟﻰ ﻋﻤﻠﯿﺔ اﻟﻀﺮب‬
‫ ﻻﺣﻘﺎ ﺳﻨﺮﻣﺰ ﻟﻤﺠﻤﻮﻋﺔ ﻛﻞ‬. a × a -1 = a -1 × a = 1 ‫ ﻟﺬا‬a -1 ‫وﺣﯿﺪ إن وﺟﺪ وﺳﻨﺮﻣﺰ ﻟﮫ ﺑﺎﻟﺮﻣﺰ‬
‫ ﯾﻤﻜﻦ ﺑﺒﺴﺎطﺔ اﻟﺒﺮھﻨﺔ ﻋﻠﻰ ان اﻟﻨﻈﺎم اﻟﺠﺒﺮي‬. R * ‫اﻟﻌﻨﺎﺻﺮ ذات اﻟﻤﻌﻜﻮس ﻓﻲ اﻟﺤﻠﻘﺔ ﺑﺎﻟﺮﻣﺰ‬
‫ ﻣﻦ اﻟﻮاﺿﺢ ان‬، (group of invertible elements of R) ‫ ( ﯾﺸﻜﻞ زﻣﺮة ﺗﺴﻤﻰ‬R * , ∙ )
‫ )ﻟﯿﺲ ﻣﻦ‬-1‫ و‬1 ‫ ﻣﺠﻤﻮﻋﺔ ﻏﯿﺮ ﺧﺎﻟﯿﺔ ﺣﯿﺚ أﻧﮭﺎ ﺗﺤﺘﻮي ﻋﻠﻰ اﻟﻌﻨﺼﺮﯾﻦ‬R * ‫اﻟﻤﺠﻤﻮﻋﺔ‬
.(‫اﻟﻀﺮوري أن ﯾﻜﻮﻧﺎ ﻣﺘﻤﺎﯾﺰﯾﻦ‬
.‫ﻓﯿﻤﺎ ﯾﻠﻲ ﺑﻌﺾ اﻷﻣﺜﻠﺔ ﻟﺘﺘﻀﺢ ﻟﺪﯾﻨﺎ اﻟﻤﻔﺎھﯿﻢ أﻋﻼه‬
Example 1.1
If Z, Q, R # denote the sets of integers, rational, and real numbers,
respectively, then (Z, +, ∙), (Q, +, ∙), (R # , +, ∙) are examples of rings;
here, + and ∙ are taken to be ordinary addition and multiplication. In each
of these cases the ring is commutative and has the integer 1 for an identity
element.
Example 1.2
Let X be a nonempty set and P(X) denote the collection of all subsets of
X. Both the systems (P(X), È, Ç ) and (P(X), Ç, È ) fail to be rings since
neither (P(X), È ) nor (P(X), Ç ) forms a group. However, (P(X), ∆) is a
commutative group, where ∆ indicates the symmetric difference
operation A ∆ B
A-B)
(
È (B-A). Since (P(X), Ç ) is clearly a
commutative semigroup and it can be proved that the operation Ç is left
distributive over ∆. The system (P(X), ∆, Ç ) is a commutative ring with
identity.
Example 1.3
(Z n , + n , × n ) is a commutative ring with identity , where
Z n = { [0], [1], …, [n-1]} is the set of integers modulo n
Example 1.4
Consider the set M 2 (R # ) of all 2x2 matrices with entries from R # . With
the usual rules of matrix addition and multiplication, (M 2 (R # ) ,+, ∙)
becomes a noncomutative ring with identity.
‫ﻣﻦ ﺧﻼل دراﺳﺘﻨﺎ ﻓﻲ اﻟﺠﺒﺮ اﻟﺨﻄﻲ) ﻓﻲ اﻟﻤﺮﺣﻠﺔ اﻷوﻟﻰ( ﻧﺴﺘﻄﯿﻊ ﺑﺴﮭﻮﻟﺔ اﻟﺘﺤﻘﻖ ﻣﻦ ﺷﺮوط‬
‫ ﻓﻜﻼ ﻋﻤﻠﯿﺘﺎ اﻟﺠﻤﻊ واﻟ ﻀﺮب ﻋﻠ ﻰ اﻟﻤ ﺼﻔﻮﻓﺎت ﺗﺠﻤﯿﻌﯿ ﺔ وﻋﻤﻠﯿ ﺔ اﻟ ﻀﺮب‬، ‫اﻟﺤﻠﻘﺔ ﻓﻲ ھﺬا اﻟﻤﺜﺎل‬
‫ ھﻲ اﻟﻌﻨﺼﺮ اﻟﺼﻔﺮي‬0 ‫ ﻛﺬﻟﻚ ﻓﺎن اﻟﻤﺼﻔﻮﻓﺔ اﻟﺘﻲ ﺟﻤﯿﻊ ﻋﻨﺎﺻﺮھﺎ‬. ‫ﺗﺘﻮزع ﻋﻠﻰ ﻋﻤﻠﯿﺔ اﻟﺠﻤﻊ‬
.[ a ij ] ‫ [ ھﻲ ﻧﻈﯿﺮ ﺟﻤﻌﻲ ﻟﻠﻤﺼﻔﻮﻓﺔ‬- a ij ] ‫ﺑﺎﻟﻨﺴﺒﺔ إﻟﻰ ﻋﻤﻠﯿﺔ اﻟﺠﻤﻊ واﻟﻤﺼﻔﻮﻓﺔ‬
‫ﻣﻦ ﻧﺎﺣﯿﺔ أﺧﺮى ﻧﺴﺘﻄﯿﻊ اﻟﺤﺼﻮل ﻋﻠﻰ أﻣﺜﻠﺔ ﺣﻠﻘﺔ ﻣﺼﻔﻮﻓﺎت ﻋﻠﻰ أي ﺣﻠﻘﺔ أﺧﺮى ﻏﯿ ﺮ ﺣﻠﻘ ﺔ‬
‫ ﺑﺸﻜﻞ ﻋﺎم‬. (n≥2) ‫ ﺑﺄي ﻋﺪد‬2 ‫اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ وﻛﺬﻟﻚ ﻧﺴﺘﺒﺪل اﻟﻌﺪد‬
(M n (R ) ,+, ∙) , is a non commutative ring with identity, where M n (R )
is the set of all n ´ n matrices with entries from R and (R, +, ∙) is an
arbitrary ring.
Example 1.5
Let X be a nonempty set and let (R, +, ∙ ) be an arbitrary ring. The
notation map(X, R) indicate the set of all mapping from X into R:
map(X, R)= { f | f: X → }
For ease of notation, let us also agree to write map(R) in place of
map(R,R).
We define addition and multiplication on map(X, R) in this way:
(f + g) (x) = f(x)+ g(x), (f ∙g)( ) f( ∙ g() ) ( Î X)
It is not difficult to verify that (map(X, R), +, ∙) is a ring. This ring is
commutative if the ring (R, +, ∙ ) is commutative . It is with identity if
(R, +, ∙ ) is with identity .
‫اﻟﻤﺤﺎﺿﺮة اﻟﺜﺎﻧﯿﺔ‬
Theorem 1.1 If R is a ring , then for any a, b, c Î R,
1.
2.
3.
4.
0a =a0 = 0,
a(-b) = (-a)b = -(ab),
(-a)(-b)= ab, and
a(b-c) = ab – ac, (b-c)a = ba – ca
Proof: ( In the class).
Note: There is one very simple ring that consists only of the additive
identity 0, with addition and multiplication given by 0+0=0, 0b0=0; this
ring is usually called the trivial ring.
Corollary Let R be a ring with identity 1 . If R is not the trivial ring,
then the elements 0 and 1 are distinct.
Proof: (In the class)
. 0 ¹ 1 ‫ﻟﻨﺘﻔﻖ ﻋﻠﻰ أن أي ﺣﻠﻘﺔ ذات ﻣﺤﺎﯾﺪ ھﻲ ﺣﻠﻘﺔ ﻏﯿﺮ ﺗﺎﻓﮭﺔ واﻟﺘﺎﻟﻲ ﻓﺎن ﺳﻨﻔﺘﺮض داﺋﻤﺎ أن‬
Definition 1.3 If R is a ring and 0 ¹ a Î R, then a is called a left (right)
zero divisor in R if there exists some b ¹ 0 in R such that ab=0 (ba=0).
A zero divisor is any element of R that is either a left or right zero
divisor.
Remarks:
1. 0 is not a zero divisor. 2. 1 is not a zero divisor. 3. any invertible
element of R is not a zero divisor.
4. If n >1 is a composite integer, that is, n= n 1 n 2 in Z ( 0< n 1 £ n 2 <n),
then the product n 1 × n n 2 =0 in Z n . So in this case the ring Z n has zero
divisors.
Theorem 1.2 A ring R is without zero divisors if and only if it satisfies
the cancellation laws for multiplication; that is, for all a,b,c in R,
ab=ac and ba=ca, where a ¹ 0 , implies that b=c.
Proof: ( In the class).
Definition 1.4 An integral domain is a commutative ring with identity
that has no zero divisors.
The ring of integers (Z, +, ∙) is the best-known example of integral
domain.
Definition 1.5 Let (R, +, ∙) be a ring and S Í R be a nonempty subset of
R. If the system (S, +, ∙) is itself a ring then (S, +, ∙) is said to be a
subring of (R, +, ∙).
. ‫اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ ﺗﺨﺘﺼﺮ ﺷﺮوط اﻟﺤﻠﻘﺔ اﻟﺠﺰﺋﯿﺔ إﻟﻰ ﺷﺮطﯿﻦ ﻣﻜﺎﻓﺌﺔ ﻟﻠﺘﻌﺮﯾﻒ‬
Theorem 1.3 Let R be a ring and f ¹ S Í R . Then S is a subring of R if
and only if
1. a, b Î S imply a-b Î S
(closure under differences),
2. a, b Î S imply ab Î S
(closure under multiplication)
Ex. 1 If S is a subring of the ring R, show that the 0 S = 0 R ; moreover, the
additive inverse of an element of the subring S is the same as its inverse
as a member of R.
Example 1.6
Every ring R has two obvious subrings, namely the set {0}, and R itself.
These two subrings are usually referred to as the trivial subrings of R; all
other subrings (if any exist) are called nontrivial. We shall use the term
proper subring to mean a subring that is different from R.
Example 1.7
The set Z e of even integers forms a subring of the ring Z of integers.
.‫ھﺬا اﻟﻤﺜﺎل ﯾﺒﯿﻦ ﻟﻨﺎ ﺣﻘﯿﻘﺔ وھﻲ أن اﻟﺤﻠﻘﺔ ذات اﻟﻤﺤﺎﯾﺪ ﯾﻤﻜﻦ أن ﺗﺤﺘﻮي ﺣﻠﻘﺔ ﺟﺰﺋﯿﺔ ﺑﺪون ﻣﺤﺎﯾﺪ‬
Example 1.8
The ring Z12 of integers modulo 12 has several nontrivial subrings,
namely {0, 6}, {0, 4, 8}, {0, 3, 6, 9} and {0, 2, 4, 6, 8, 10}.