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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}.