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Exercise 2.46.1 Let R be a commutative ring. Suppose that every nonzero element in R is invertible. Prove that R cannot have zero-divisors, and hence, R must a field. Give an example of a commutative ring to show that, conversely, if R is commutative and has no zero-divisors, then all nonzero elements need not be invertible.

Exercise 2.46.1 Let R be a commutative ring. Suppose that every nonzero element in R is invertible. Prove that R cannot have zero-divisors, and hence, R must a field. Give an example of a commutative ring to show that, conversely, if R is commutative and has no zero-divisors, then all nonzero elements need not be invertible.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Exercise 2.46.1** Let \( R \) be a commutative ring. Suppose that every nonzero element in \( R \) is invertible. Prove that \( R \) cannot have zero-divisors, and hence, \( R \) must be a field. Give an example of a commutative ring to show that, conversely, if \( R \) is commutative and has no zero-divisors, then all nonzero elements need not be invertible.
Transcribed Image Text:**Exercise 2.46.1** Let \( R \) be a commutative ring. Suppose that every nonzero element in \( R \) is invertible. Prove that \( R \) cannot have zero-divisors, and hence, \( R \) must be a field. Give an example of a commutative ring to show that, conversely, if \( R \) is commutative and has no zero-divisors, then all nonzero elements need not be invertible.
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