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J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1 If a 1 and a² = 1, then a + 1 and a 1 are divisors of zero. #2 If ab is a divisor of zero, then a or b is a divisor of zero. 3 In a commutative ring with unity, a divisor of zero cannot be invertible.

J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1 If a 1 and a² = 1, then a + 1 and a 1 are divisors of zero. #2 If ab is a divisor of zero, then a or b is a divisor of zero. 3 In a commutative ring with unity, a divisor of zero cannot 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
Problem 1RQ
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J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1 If a 1 and a² = 1, then a + 1 and a 1 are divisors of zero. #2 If ab is a divisor of zero, then a or b is a divisor of zero. 3 In a commutative ring with unity, a divisor of zero cannot be invertible. 4 Suppose ab #0 in a commutative ring. If either a or is a divisor of zero, so is ab. 5 Suppose a is neither 0 nor a divisor of zero. If ab = ac, then b = c. 6 Ax B always has divisors of zero.
Transcribed Image Text:J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1 If a 1 and a² = 1, then a + 1 and a 1 are divisors of zero. #2 If ab is a divisor of zero, then a or b is a divisor of zero. 3 In a commutative ring with unity, a divisor of zero cannot be invertible. 4 Suppose ab #0 in a commutative ring. If either a or is a divisor of zero, so is ab. 5 Suppose a is neither 0 nor a divisor of zero. If ab = ac, then b = c. 6 Ax B always has divisors of zero.
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