Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 13, Problem 9E
Find elements a, b, and c in the ring
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Give an example where a and b are not zero divisors in a ring R, but the sum a + b is a zero divisor.
Find elements a, b, and c in the ring Z ⨁ Z ⨁ Z such that ab, ac, andbc are zero-divisors but abc is not a zero-divisor.
Chapter 13 Solutions
Contemporary Abstract Algebra
Ch. 13 - Prob. 1ECh. 13 - Show that a commutative ring with the cancellation...Ch. 13 - Show that every nonzero element of Zn is a unit or...Ch. 13 - Find a nonzero element in a ring that is neither a...Ch. 13 - Let R be a finite commutative ring with unity....Ch. 13 - Find elements a, b, and c in the ring ZZZ such...Ch. 13 - Describe all zero-divisors and units of ZQZ .Ch. 13 - Let a belong to a ring R with unity and suppose...Ch. 13 - Show that 0 is the only nilpotent element in an...Ch. 13 - Prob. 18E
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- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forward
- 28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.arrow_forward43. Let . a. Show that is a noncommutative subring of . b. Find the unity element, if it exists.arrow_forwardGiven that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.arrow_forward
- Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R is a commutative ring with unity with respect to matrix addition and multiplication. Answer the following questions and give a reason for any negative answers. Is 12 an integral domain? Is R a field?arrow_forwardSuppose that a,b, and c are elements of a ring R such that ab=ac. Prove that is a has a multiplicative inverse, then b=c.arrow_forward22. Define a new operation of addition in by and a new multiplication in by. a. Is a commutative ring with respect to these operations? b. Find the unity, if one exists.arrow_forward
- 42. Let . a. Show that is a commutative subring of. b. Find the unity, if one exists. c. Describe the units in, if any.arrow_forwardLet [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
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