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:اﻟﻤﺤﺎﺿﺮة اﻟﺜﺎﻟﺜﺔ .ﻓﯿﻤﺎ ﯾﻠﻲ ﺳﻨﻌﺮف ﻣﺮﻛﺰ اﻟﺤﻠﻘﺔ ﺛﻢ ﻧﺜﺒﺖ أﻧﮭﺎ ﺣﻠﻘﺔ ﺟﺰﺋﯿﺔ ﻣﻦ اﻟﺤﻠﻘﺔ اﻷﺻﻠﯿﺔ Definition 1.6 Let R be a ring, then cent R is the set of all elements of R that commute with every element of R, that is cent R = { a Î R| ar =ra for all r Î R} .cent R= R ﺗﻜﻮن اﺑﺪاﻟﯿﺔ إذا وﻓﻘﻂ إذاR ﻣﻦ اﻟﻮاﺿﺢ أن اﻟﺤﻠﻘﺔ Theorem 1.4 For any ring R, cent R is a subring of R. Proof: 0 Î cent R , since a0=0a =0 "a Î R, which implies cent R ¹ f . Let a, b Î cent R, then (a- b)r = ar – br = ra – rb = r(a – b). Hence a – b Î cent R. Also, (ab)r = a(br) =a(rb)= (ar)b=(ra)b=r(ab), which implies ab Î cent R too. Therefore by Theorem 1.3 cent R is a subring. ﻻﺣﻈﻨﺎ ﻓﻲ ﻣﺜ ﺎل ﺳ ﺎﺑﻖ اﻧ ﮫ ﻣ ﻦ اﻟﻤﻤﻜ ﻦ أن ﺗﻜ ﻮن ﺣﻠﻘ ﺔ ذات ﻋﻨ ﺼﺮ ﻣﺤﺎﯾ ﺪ وﻟ ﺪﯾﮭﺎ ﺣﻠﻘ ﺔ ﺟﺰﺋﯿ ﺔ : وﯾﻨﺒﻐﻲ أن ﻧﻌﻠﻢ أن اﻟﺤﺎﻻت اﻟﺘﺎﻟﯿﺔ ﻣﻤﻜﻨﺔ أﯾﻀﺎ،ﺑﺪون ﻋﻨﺼﺮ ﻣﺤﺎﯾﺪ . ﺣﻠﻘﺔ ﺑﺪون ﻣﺤﺎﯾﺪ وﺗﺤﻮي ﺣﻠﻘﺔ ﺟﺰﺋﯿﺔ ﺑﻤﺤﺎﯾﺪ-1 . ﺣﻠﻘﺔ ﺑﻤﺤﺎﯾﺪ ﺗﺤﻮي ﺣﻠﻘﺔ ﺟﺰﺋﯿﺔ ﺑﻤﺤﺎﯾﺪ ﯾﺨﺘﻠﻒ ﻋﻦ ﻣﺤﺎﯾﺪ اﻟﺤﻠﻘﺔ اﻷﺻﻠﯿﺔ-2 وﻓﯿﻤ ﺎ ﯾﻠ ﻲ.ﻓﻲ ﻛﻼ اﻟﺤﺎﻟﺘﯿﻦ أﻋﻼه ﯾﻜﻮن ﻣﺤﺎﯾﺪ اﻟﺤﻠﻘ ﺔ اﻟﺠﺰﺋﯿ ﺔ ﻗﺎﺳ ﻢ ﻟﻠ ﺼﻔﺮ ﻓ ﻲ اﻟﺤﻠﻘ ﺔ اﻷﺻ ﻠﯿﺔ .ﺑﺮھﺎن ھﺬه اﻟﺤﻘﯿﻘﺔ Let 1¢ ¹ 0 denote the identity element of the subring S; we assume further that 1¢ does not act as an identity for the whole ring R. Accordingly, some element aÎ R exists for which a1¢ ¹ a . It is clear that (a 1¢ ) 1¢ = a(1¢ 1¢ )=a 1¢ . or ( a 1¢ - a) 1¢ =0. Since neither a1¢ -a nor 1¢ is zero, the ring R has zero divisors, and in particular, 1¢ is a zero divisor. ﻓﯿﻤﺎ ﯾﻠﻲ ﻣﺜﺎل ﻋﻦ ﺣﻠﻘﺔ وﺣﻠﻘﺔ ﺟﺰﺋﯿﺔ ﻛﻼ ﻣﻨﮭﻤﺎ ذات ﻋﻨﺼﺮ ﻣﺤﺎﯾﺪ وﻟﻜﻦ ﻣﺤﺎﯾﺪ اﻟﺤﻠﻘﺔ ﯾﺨﺘﻠﻒ ﻋﻦ .ﻣﺤﺎﯾﺪ اﻟﺤﻠﻘﺔ اﻟﺠﺰﺋﯿﺔ Example 1.9 Consider the set R= Z ´ Z, consisting of ordered pairs of integers. One converts R into a ring by defining addition and multiplication componentwise: (a, b) + (c, d) = (a + c, b +d); (a, b)(c, d) = (ac, bd) A routine calculation will show that Z ´ {0} = {(a, 0)| a Î Z} forms a subring with identity element (1, 0). While the identity of the entire ring R is (1, 1). Also note that (1, 0)(0, 1)= (0, 0), that is, (1, 0) (the identity of the subring) is a zero divisor of R. .( ﻓﯿﻤﺎ ﯾﻠﻲ ﻧﺘﻔﻖ ﻋﻠﻰ رﻣﺰ اﻟﻘﻮة ﻟﻌﻨﺎﺻﺮ اﻟﺤﻠﻘﺔ ) اﻟﻀﺮب اﻟﻤﺘﻜﺮر ﻟﻨﻔﺲ اﻟﻌﻨﺼﺮ If R is an arbitrary ring and n a positive integer, then the nth power a of an element a Î R is defined by the inductive conditions a 1 = a and a n = a n -1a . From this the usual laws of exponents follow at once: a n a m = a n + m , (a n ) m = a nm ( n, m Î Z + ) Observe that if two elements n, mÎ R happen to commute, then so do all powers of a and b, whence (ab) n = a n b n for each positive integer n. ﯾﻤﻜﻦ اﻋﺘﻤﺎد اﻟﻘﻮىa ﻛﺬﻟﻚ ﻓﻲ ﺣﺎﻟﺔ وﺟﻮد اﻟﻤﺤﺎﯾﺪ ﻓﻲ اﻟﺤﻠﻘﺔ ووﺟﻮد اﻟﻨﻈﯿﺮ اﻟﻀﺮﺑﻲ ﻟﻠﻌﻨﺼﺮ : أﯾﻀﺎ وﻛﻤﺎ ﯾﻠﻲa اﻟﺴﺎﻟﺒﺔ ﻟﻠﻌﻨﺼﺮ 0 -n -1 n a = (a ) , where n>0. With the definition a = 1 the symbol a n now has a well-defined meaning foe every integer n – at least when the attention is restricted to invertible elements. ﺑﺎﻟﻨﺴﺒﺔ إﻟﻰ ﻋﻤﻠﯿﺔ اﻟﺠﻤﻊ ﯾﻤﻜﻦ أﯾﻀﺎ أن ﻧﻌﺒﺮ ﻋﻦ ﺗﻜﺮار اﻟﺠﻤﻊ ﻟﻌﻨﺼﺮ واﺣﺪ ﺑﺎﻟﻤﻀﺎﻋﻔﺎت :اﻟﺼﺤﯿﺤﺔ وﻛﻤﺎ ﯾﻠﻲ 1a=a, and na = (n-1) a + a, when n>1 0a= 0 and (-n) a= -(na). :ﻛﻤﺎ ﯾﻤﻜﻦ ﺑﺴﮭﻮﻟﺔ ﺑﺮھﻨﺔ اﻟﻌﺒﺎرات اﻟﺘﺎﻟﯿﺔ (n+ m) a = na + ma, (nm)a= n(ma), n(a +b) = na + nb, n(ab) = (na) b = a(nb), and (na)(mb)= (nm)(ab). ﯾﻤﻜﻦ أن ﻻ ﯾﻜﻮنn ﺣﯿﺚ أن اﻟﻌﺪد، ﻻﺗﻤﺜﻞ ﺿﺮب ﻓﻲ اﻟﺤﻠﻘﺔna ﯾﻨﺒﻐﻲ أن ﻧﻔﮭﻢ أن اﻟﺼﯿﻐﺔ . وإﻧﻤﺎ ھﻮ رﻣﺰ)ﻣﺘﻔﻖ ﻋﻠﯿﮫ( ﻟﺠﻤﻊ ﻣﺎ،ﻋﻨﺼﺮ ﻓﻲ اﻟﺤﻠﻘﺔ .وأﻻن ﻧﻌﺮف ﻣﻤﯿﺰ اﻟﺤﻠﻘﺔ Definition 1.6 Let R be an arbitrary ring. If there exists a positive integer n such that na= 0 for all aÎ R, then the smallest positive integer with this property is called the characteristic of the ring . If no such positive integer exists (that is, n=0 is the only integer for which na=0 for all a in R), then R is said to be of characteristic zero. We shall write char R for the characteristic of R. . اﻟﻨﺴﺒﯿﺔ واﻟﺤﻘﯿﻘﯿﺔ ھﻲ أﻣﺜﻠﺔ واﺿﺤﺔ ﻟﺤﻠﻘﺎت ﻣﻤﯿﺰھﺎ ﺻﻔﺮا، ﺣﻠﻘﺎت اﻷﻋﺪاد اﻟﺼﺤﯿﺤﺔ ﺣﯿﺚ ان2 ( ھﻲ ﺣﻠﻘﺔ ﻣﻤﯿﺰھﺎ1.2 ) ﻣﺜﺎلP(X) ﻣﻦ ﻧﺎﺣﯿﺔ أﺧﺮى اﻟﺤﻠﻘﺔ 2A = A ∆ A = f for every subset A of X. ﻟﻜﻦ ﻓﻲ اﻟﺤﻠﻘﺎت ذات،ﺑﺎﻟﺮﻏﻢ ﻣﻦ ان ﺗﻌﺮﯾﻒ اﻟﻤﻤﯿﺰ ﯾﺘﻄﻠﺐ ﺗﺤﻘﻖ ﺷﺮط ﻟﻜﻞ ﻋﻨﺎﺻﺮ اﻟﺤﻠﻘﺔ . ﻻﺣﻆ اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ.اﻟﻤﺤﺎﯾﺪ ﯾﻜﻔﻲ ﺗﺤﻘﻖ اﻟﺸﺮط ﻓﻲ اﻟﻤﺤﺎﯾﺪ ﻓﻘﻂ n Theorem 1.5 If R is a ring with identity 1, then R has characteristic n>0 if and only if n is the least positive integer for which n1=0. Proof: In the class. Corollary 1 In an integral domain R, all the nonzero elements have the same additive order; this order is the characteristic of the domain when char R> 0 and is infinite when char R = 0. Corollary 2 An integral domain R has positive characteristic if and only if na = 0 for some 0 ¹ a Î R and some integer nÎ Z + . Theorem 1.6 The characteristic of an integral domain is either zero or a prime number. Proof: Assume that char R =n> 0 and n is not prime. Then n= n 1 n 2 , with 1< n 1 £ n 2 < n. It follows that 0 = n 1 = ( n 1 n 2 ) 1 = ( n 1 n 2 ) 1 2 = (n 1 1)( n 2 1). By assumption , R is without zero divisors, so either n 1 = 0 or n 2 = 0. Since both n 1 and n 2 are less than n, this is a contradiction. Therefore char R must be prime. Corollary If R is a finite integral domain, then char R = p, where p is a prime. Remark. Let R be a ring with identity and consider the set Z1= { n1| nÎ Z}, then 1. Z1 forms a (commutative) ring with identity. 2. The order of the additive cyclic group (Z1, +) = char R. 3. If R is an integral domain, then Z1 is a subdomain of R . It is the smallest subdomain of R. 4. If char R=p, where p is a prime, then Z1 is a field. Proof: 1. n1 – m1 = (n – m) 1, (n1)(m1) = (n m)1 and (1) 1= 1. 2. Clear. 3. (n1)(m1) = 0 implies (n m)1 = 0 implies n m =0 implies n =0 or m = 0 implies n1= 0 or m1= 0. If S is any subdomain of R, then 1S = 1R ( otherwise 1S is a zero divisor of R), hence 1Î S, that is Z1 Í S. 4. Let char R=p, where p is a prime and R is an integral domain, then it is enough to prove that each nonzero element of Z1 is invertible. Since Z1 is an additive cyclic group of order p, it consists of p distinct elements, namely, the p sums n1, where n = 0, 1, …, p – 1. Let n1 ¹ 0 ( 0 < n < p). Since n and p are relatively prime, integers r and s exist for which r p + s n =1. But then 1 = (r p + s n)1 = r(p1) + (s1)( n1). As p1 = 0, we obtain the equation 1 = (s1)(n1), so that s1 serves as the multiplicative inverse of n1 in Z1. اﻟﻤﺤﺎﺿﺮة اﻟﺮاﺑﻌﺔ Definition 1.7 Let R and R ¢ be two rings . By a (ring) homomorphism from R into R ¢ is meant a function f : R→ R ¢ such that f (a + b) = f ( a) + f ( b), f (a ∙ b) = f ( a) ∙ f (b) for every pair of elements a, bÎ R. A homomorphism that is also a one to one mapping is called an isomorphism. ﯾﻨﺒﻐﻲ اﻻﻧﺘﺒﺎه إﻟﻰ أن ﻋﻤﻠﯿﺘﺎ اﻟﺠﻤﻊ واﻟﻀﺮب اﻟﻈﺎھﺮة ﻋﻠﻰ اﻟﯿﺴﺎر ﻓﻲ اﻟﺘﻌﺮﯾﻒ أﻋﻼه ﺗﺨﺺ . R ¢ واﻟﻈﺎھﺮة ﻋﻠﻰ اﻟﯿﻤﯿﻦ ﺗﺨﺺ اﻟﺤﻠﻘﺔR اﻟﺤﻠﻘﺔ If f is homomorphism from R into R ¢ , then f ( R) is called the homomorphic image of R. When R = R ¢ , we speak of f as a homomorphism of R into itself or an endomorphism of R. Likewise, an isomorphism of the ring R onto itself is termed an automorphism of R. Example 1.10 Let R and R ¢ be arbitrary rings and f: R→ R ¢ be the function that sends each element of R onto the zero element of R ¢ . Example 1.11 Consider ring Z of integers and the ring Z n of integers modulo n. Define f : Z ® Z n by taking f ( a) = [a]; that is, map each integer into the congruence class containing it. Example 1.12 In the ring map(X, R), define t a to be the function that assigns to each f Î map(X, R) its value at a fixed element a Î X; in other words, t a ( f)= f (a). Then is a homomorphism from map(X, R) into R, known as the evaluation homomorphism at a. .وأﻻن ﻧﺴﺘﻨﺘﺞ ﺑﻌﺾ اﻟﺨﻮاص اﻷوﻟﯿﺔ ﻟﻠﺘﺸﺎﻛﻞ ﻣﻦ ﺧﻼل اﻟﺘﻌﺮﯾﻒ Theorem 1.7 If f is a homomorphism from the ring R into the ring R ¢ , the following hold: 1. f (0) = 0 and 2. f (- a ) = - f (a) for all aÎ R. If, in addition, R and R ¢ are both rings with identity and f ( R)= R ¢ , then 3. f (1) = 1 and 4. f ( a -1 ) = f (a ) -1 for each invertible element aÎ R. Proof: In the class. Theorem 1.8 Let f be a homomorphism from the ring R into the ring R ¢ . Then a) for each subring S of R, f(S) is a subring of R ¢ ; and b) for each subring S¢ of R ¢ , f -1 ( S¢ ) is a subring of R. Proof: In the class. In agreement with our previous use of the term, two rings R and R ¢ are said to be isomorphic if there exists an isomorphism from R onto R ¢ ; we indicate this by writing R @ R ¢ . Definition 1.8 A ring R is said to be imbedded in a ring R ¢ if there exists some subring S¢ of R ¢ such that R @ S¢ . In general, if a ring R is imbedded in a ring R ¢ , then R ¢ is referred to as an extension of R, and we say that R can be extended to R ¢ . Theorem 1.9 (Dorroh Extension Theorem) Any ring R can be imbedded in a ring with identity. Proof: Consider the Cartesian product R ´ Z, where R ´ Z= {(r, n)| r Î R; nÎ Z}. If addition and multiplication are defined by (a, n) + (b, m) = (a + b, n + m), (a, n)(b, m) = (ab + ma + nb, nm), then it is a simple matter to verify that R ´ Z forms a ring. Note that this system has a multiplicative identity, namely, (0, 1). Next, consider the subset R ´ {0} of R ´ Z consisting of all pairs of the form (a, 0), it is a subring and isomorphic to R under the mapping f: R→R ´ {0} defined by f(a)= (a, 0). This process of extension therefore imbeds R in R ´ Z, a ring with identity. Problems 1 1. Consider the set Z of integers with the usual operations. If a new addition and multiplication are defined by a * b= a+b+1, a o b= a+ b+ ab, prove that the system (Z, * , o ) is a commutative ring with identity. 2. Establish the following statements: a) The group Z * 5 of invertible elements of the ring Z 5 is isomorphic to the cyclic group Z 4 . b) The group Z * 9 of invertible elements of the ring Z 9 is cyclic with generators 2 and 5. 3. In a ring R with identity, prove that each of the following is true: a) The identity element for multiplication is unique. b) If a Î R has a multiplicative inverse, then a -1 is unique. c) If an element aÎ R is invertible, then so also is –a. d) No zero divisor of R can possess a multiplicative inverse. 4. a) If the set X contains more than one element, show that every nonempty proper subset of X is a zero divisor in the ring P(X). c) For the ring map R # , verify that the function f(x) = x + |x| and g(x) = x -|x| are both zero divisors. 5. An element a of a ring R is said to be idempotent if a 2 = a and nilpotent if a n = 0 for some n Î Z + . Show that a) a nonzero idempotent element cannot be nilpotent; b) every nonzero nilpotent element is a zero divisor. c) If R is an integral domain, the only nilpotent element is the zero element; d) If R is an integral domain, the identity element is the only nonzero idempotent element.