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6.6 Rings and fields 6.6.1 Rings Definition 21: A ring is an Abelian group [R, +] with an additional associative binary operation (denoted ·) such that for all a, b, c R, (1) a · (b + c) = a · b + a · c, (2) (b + c) · a = b · a + c · a. We write 0 R for the identity element of the group [R, +]. For a R, we write -a for the additive inverse of a. Remark: Observe that the addition operation is always commutative while the multiplication need not be. Observe that there need not be inverses for multiplication.
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Example: The sets Z,Q, with the usual operations of multiplication and addition form rings, [Z;+, ],[Q;+, ] are rings Let M={(a ij ) n n |a ij is real number}, Then [M;+, ]is a ring Example: S ,[P(S); ,∩] , Commutative ring
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Definition 23: A ring R is a commutative ring if ab = ba for all a, b R. A ring R is an unitary ring if there is 1 R such that 1a = a1 = a for all a R. Such an element is called a multiplicative identity.
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Example: If R is a ring, then R[x] denotes the set of polynomials with coefficients in R. We shall not give a formal definition of this set, but it can be thought of as: R[x] = {a 0 + a 1 x + a 2 x 2 + …+ a n x n |n Z +, a i R}. Multiplication and addition are defined in the usual manner; if then One then has to check that these operations define a ring. The ring is called polynomial ring.
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Theorem 6.26: Let R be a commutative ring. Then for all a,b R, where n Z +.
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1. Identity of ring and zero of ring Theorem 6.27: Let [R;+,*] be a ring. Then the following results hold. (1)a*0=0*a=0 for a R (2)a*(-b)=(-a)*b=-(a*b) for a,b R (3)(-a)*(-b)=a*b for a,b R Let 1 be identity about *. Then (4)(-1)*a=-a for a R (5)(-1)*(-1)=1
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1:Identity of ring 0:zero of ring
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[M 2,2 (Z);+, ] is an unitary ring Zero of ring (0) 2 2, Identity of ring is
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2. Zero-divistors Definition 23: If a 0 is an element of a ring R for which there exists b 0 such that ab=0(ba=0), then a (b) is called a left(right) zero-divistor in R. Let S={1,2} , is zero element of ring [P(S); ,∩]
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6.6.2 Integral domains, division rings and fields Definition 24: A commutative ring is an integral domain if there are no zero- divisors. [P(S); ,∩] and [M;+, ] are not integral domain, [Z;+, ] is an integral domain Theorem 6.28: Let R be a commutative ring. Then R is an integral domain iff. for any a, b, c R if a 0 and ab=ac, then b=c. Proof: 1)Suppose that R is an integral domain. If ab = ac, then ab - ac=0
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2)R is a commutative ring, and for any a, b, c R if a 0 and ab=ac, then b=c. Prove: R is an integral domain
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Let [R;+;*] be a ring with identity element 1. If 1=0, then for a R, a=a*1=a*0=0. Hence R has only one element, In other words, If |R|>1, then 1 0.
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Definition 25: A ring is a division ring if the non- zero elements form a group under multiplication. If R is a division ring, then |R| 2. Ring R has identity, and any non-zero element exists inverse element under multiplication. Definition 26: A field is a commutative division ring. [Z;+, ]is a integral domain, but it is not division ring and field [Q;+, ], [R;+, ]and[C;+, ] are field
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Let [F;+,*] be a algebraic system, and |F| 2, (1)[F;+]is a Abelian group (2)[F-{0};*] is a Abelian group (3)For a,b,c F, a*(b+c)=(a*b)+(a*c)
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Theorem 6.29: Any Field is an integral domain Let [F;+,*] be a field. Then F is a commutative ring. If a,b F-{0}, s.t. a*b =0 。 [Z;+, ] is an integral domain. But it is not a field
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Next: fields, Subring, Ideal and Quotient ring,Ring homomorphism Exercise:P381(Sixth) OR P367(Fifth) 7,8, 16,17,20 1.Let X be any non-empty set. Show that [P(X); ∪, ∩] is not a ring. 2. Let Z[i] = {a + bi| a, b Z}. (1)Show that Z[i] is a commutative ring and find its units. (2)Is Z[i] a field? Why? 3.Show that Q[i] = {a + bi | a, b Q} is a field.
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