Lemma: Let CR+.) be aring then centR=R iffRis0Commutative ring. (H-W)

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Answered: Lemma: Let CR+.) be aring then centR=R…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 55RE

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[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]
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Show that (Z,+,.) Is embedded in (Q,+,.).?? Question from rings and fields subject
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(R, +, .) (N, +, .) (R, +, .) (Z, +, .) | (Z6, +, .) (M3x3(IR), +, .) Ring Has identity Commutative Every nonzero element has inverse Has no zero divisor Division ring Integral domain Field Char (R)
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Qa: Prove that the ring of rational numbers (Q, +,.) is division ring
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Check if the solution is correct. Correct the mistakes.If x4=x for any x in R, prove that R is commutativeLet a,b be official elements of the ring R. Then:(a+b)4 = a4 + 4a3 b + 6a2b2 + 4ab3 +b4Note that each term of this number is equal to either a or b (or their product), according to the condition x^4=x for any x in R.(a+b)4 = a + bNow consider the expression (b+a)4 :(b+a)4 = b + aHence we have:a + b = (a+b)4 = (b+a)4 = b + aThis means that for any elements a and b in the ring R, the stability of commutativity increases, that is, R is a commutative ring.
(Q1)Find all idempotent elements and nilpotent elements of the following rings 2- (Z/(3),+.) + (C. +,.)

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Lemma: Let CR+.) be aring then centR=R iff Ris 0 Commutative ring. (H-W)