Given a well-ordered integral domain with unity e ED and positive elements D, and the mapping 0: Z→ D defined by 0(n)=ne Vn e Z. (a) Prove that e is the least positive element in D. (b) True or false? Justify your answer: The image set 0(Z) is a subring of D. (c) True or false? Justify your answer: If an element de D is not in 0(Z), then -d0(Z). :D\0(Z). Prove that SP, the set of (d) Suppose that the subset S CD is defined as S := positive elements of S. has no least element. What does this say about S? (e) Hence show that Z~ D, i.e., that any well-ordered integral domain is isomorphic to the ring of integers.
Given a well-ordered integral domain with unity e ED and positive elements D, and the mapping 0: Z→ D defined by 0(n)=ne Vn e Z. (a) Prove that e is the least positive element in D. (b) True or false? Justify your answer: The image set 0(Z) is a subring of D. (c) True or false? Justify your answer: If an element de D is not in 0(Z), then -d0(Z). :D\0(Z). Prove that SP, the set of (d) Suppose that the subset S CD is defined as S := positive elements of S. has no least element. What does this say about S? (e) Hence show that Z~ D, i.e., that any well-ordered integral domain is isomorphic to the ring of integers.
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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Transcribed Image Text:Given a well-ordered integral domain < D. +, >
with unity e ED and positive elements D, and the mapping 0: Z→→ D defined by
0(n)=ne Vn e Z.
(a) Prove that e is the least positive element in D.
(b) True or false? Justify your answer: The image set 0(Z) is a subring of D.
(c) True or false? Justify your answer: If an element de D is not in 0(Z), then -d 0(Z).
(d) Suppose that the subset S CD is defined as S = D\0(Z). Prove that SP, the set of
positive elements of S. has no least element. What does this say about S?
(e) Hence show that Z~ D, i.e., that any well-ordered integral domain is isomorphic to the
ring of integers.
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VIEWStep 2: Prove that e is the least positive element in D
VIEWStep 3: Determine if the image set θ(ℤ) is a subring of D.
VIEWStep 4: Determine if an element d ∈ D is not in θ(ℤ) then -d ∉ θ(ℤ).
VIEWStep 5: Proving that S^p has no least element.
VIEWStep 6: Showing that ℤ is isomorphic to D.
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