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
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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), the
ideal of R[x] generated by the element ax – 1R. Show R[x]/I ≈ S¹R, where
S¹R = {:r € R, s ¤ Sa}
a
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98ddc1e7-1afc-4f0e-b623-cf5e11e78fa9%2F0262dd92-a5ac-4bc4-a387-e5141e51503b%2F946z1jf_processed.png&w=3840&q=75)
Transcribed Image Text: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), the
ideal of R[x] generated by the element ax – 1R. Show R[x]/I ≈ S¹R, where
S¹R = {:r € R, s ¤ Sa}
a
=
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