Solution please. Should correct,no ai answer Let R be a commutative ring with 1Ŕ and let a € R− {0R} be not nilpotent.Define Sa {an: ne Zo}. Let IC R[x] be defined by I = (ax - 1R), theideal of R[x] generated by the element ax – 1R. Show R[x]/I ≈ S¹R, whereS¹R = {:r € R, s ¤ Sa}a=

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Answered: Let R be a commutative ring with 1Ŕ and…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 24E: 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set...

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Let R be a commutative ring with 1Ŕ and let a € R - {OR} be not nilpotent. Define Sa = {ann € Z≥o}. Let I ≤ R[x] be defined by I = (ax – 1Ŕ), the ideal of R[x] generated by the element ax - 1R. Show R[x]/I ≈ S¹R, where S¹R={:re R, 8 € Sa} S