   # a. Give an example where a and b are not zero divisors in a ring R , but the sum a + b is a zero divisor. Give an example where a and b are zero divisors in a ring R with a + b ≠ 0 , and a + b is not a zero divisor. Prove that the set of all elements in a ring R that are not zero divisors is closed under multiplication. ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

#### Solutions

Chapter
Section ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 5.2, Problem 19E
Textbook Problem
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## a. Give an example where a and b are not zero divisors in a ring R , but the sum a   +   b is a zero divisor.Give an example where a and b are zero divisors in a ring R with a   +   b   ≠   0 , and a   +   b is not a zero divisor.Prove that the set of all elements in a ring R that are not zero divisors is closed under multiplication.

(a)

To determine

An example where a and b are not zero divisors in a ring R, but the sum a+b is a zero divisor.

### Explanation of Solution

Let R=8={,,,,,,,} be a ring.

For  and  in 8, there doesn’t exist any [a] element in 8.

Such that [a]= or [a]=.

 and  are not zero divisors in a ring 8

(b)

To determine

An example where a and b are zero divisors in a ring R with a+b0, and a+b is not a zero divisor.

(c)

To determine

To prove: The set of all elements in a ring, R that are not zero divisors, is closed under multiplication.

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