1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes a claim, no matter how trivial. It is available in a Word document for your convenience. *** Let R be a commutative ring with unity and I an ideal of R. The quotient ring R/I is an integral domain if and only if I is prime.** (=) Assume I is a prime ideal of R. Assume a, b ER/I such that ab = .. a = 1 + a2 : b = 1 + b2 for some b2 E R. :: (1 + az)(I + b2) = 1 + OR :I + azb, = 1 + OR : azb, – OR E I . azbz E I : az El or b, € I Wlog az E I = OR/I- for some a, ER. az – OR E I :I+ az = I +0R Or/I : R/l is an integral domain. :: a =
1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes a claim, no matter how trivial. It is available in a Word document for your convenience. *** Let R be a commutative ring with unity and I an ideal of R. The quotient ring R/I is an integral domain if and only if I is prime.** (=) Assume I is a prime ideal of R. Assume a, b ER/I such that ab = .. a = 1 + a2 : b = 1 + b2 for some b2 E R. :: (1 + az)(I + b2) = 1 + OR :I + azb, = 1 + OR : azb, – OR E I . azbz E I : az El or b, € I Wlog az E I = OR/I- for some a, ER. az – OR E I :I+ az = I +0R Or/I : R/l is an integral domain. :: a =
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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Question
![1. Consider the proof of the backwards direction of the theorem below. Justify each step that
makes a claim, no matter how trivial. It is available in a Word document for your convenience.
*** Let R be a commutative ring with unity and I an ideal of R. The quotient ring R/I is an integral
domain if and only if I is prime.**
(=) Assume I is a prime ideal of R.
Assume a, b ER/I such that ab =
= OR/I.
. a = 1 + a2 for some a, E R.
: b = 1 + b2 for some b, E R.
: (1 + az)(I + b2) = 1 + OR
: I + azb, = 1 + OR
: azb, – OR E I
: azb, E I
: az El or b, e I
Wlog az E I
az – OR E I
:I+ az = I +l0R
Or/I
: R/l is an integral domain.
:: a =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30e714d6-c3d6-4fe9-9c10-b4b8c64f125f%2F944b749a-9028-45ab-aa6b-70ab5c507268%2Fkp9st2k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider the proof of the backwards direction of the theorem below. Justify each step that
makes a claim, no matter how trivial. It is available in a Word document for your convenience.
*** Let R be a commutative ring with unity and I an ideal of R. The quotient ring R/I is an integral
domain if and only if I is prime.**
(=) Assume I is a prime ideal of R.
Assume a, b ER/I such that ab =
= OR/I.
. a = 1 + a2 for some a, E R.
: b = 1 + b2 for some b, E R.
: (1 + az)(I + b2) = 1 + OR
: I + azb, = 1 + OR
: azb, – OR E I
: azb, E I
: az El or b, e I
Wlog az E I
az – OR E I
:I+ az = I +l0R
Or/I
: R/l is an integral domain.
:: a =
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