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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
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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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 =
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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