Let R be a commutative ring. Given elements a, b ∈ R, we say that a divides b, and write a | b, if there exists an element d ∈ R such that b = da. This defines a relation | on the set R. (a) Prove that | is reflexive and transitive. (b) Prove that | fails to be symmetric. (c) Prove that | restricted to the set R∗ = R \ {0R} of nonzero elements is symmetri

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

Let R be a commutative ring. Given elements a, b ∈ R, we say that a divides b, and
write a | b, if there exists an element d ∈ R such that b = da. This defines a relation |
on the set R.
(a) Prove that | is reflexive and transitive.
(b) Prove that | fails to be symmetric.
(c) Prove that | restricted to the set R∗ = R \ {0R} of nonzero elements is symmetric
if and only if R is a field.
(d) Give an example of a commutative ring R for which | is anti-symmetric. (Hint:
Suppose R is anti-symmetric, and consider the element −1R.)

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