1. State the Cancellation Axiom for the multiplication. 2. Show, justifying each step, that the Cancellation Axiom holds if and only if R has no zero-divisors, i.e., if a, b e R and a xb= 0, then a = 0 or b = 0.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 26E: Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.
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Let R be a commutative ring with 1, i.e, on R we have two binary operations (addition and multiplication) and the Rules of Arithmetic (A1) - (A5) and (M1) - (M4) hold. Please solve the attached questions. Thank you!

1. State the Cancellation Axiom for the multiplication.
2. Show, justifying each step, that the Cancellation Axiom holds if and
only if R has no zero-divisors, i.e., if a, b e R and a xb= 0, then a = 0
or b = 0.
Transcribed Image Text:1. State the Cancellation Axiom for the multiplication. 2. Show, justifying each step, that the Cancellation Axiom holds if and only if R has no zero-divisors, i.e., if a, b e R and a xb= 0, then a = 0 or b = 0.
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