Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 13, Problem 19E
Let a andb be idempotents in a commutative ring. Show that eachof the following is also an idempotent:
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Let a and b be idempotents in a commutative ring. Show that eachof the following is also an idempotent: ab, a - ab, a + b - ab,a + b - 2ab.
Give an example where a and b are not zero divisors in a ring R, but the sum a + b is a zero divisor.
Suppose that a and b belong to a commutative ring and ab is a zero-divisor.Show that either a or b is 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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- 19. Find a specific example of two elements and in a ring such that and .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_forward35. The addition table and part of the multiplication table for the ring are given in Figure . Use the distributive laws to complete the multiplication table. Figurearrow_forward
- The addition table and part of the multiplication table for the ring R={ a,b,c } are given in Figure 5.1. Use the distributive laws to complete the multiplication table. Figure 5.1 +abcaabcbbcaccab abcaaaabaccaarrow_forwardAn element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().arrow_forwardLet 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_forward
- 22. 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_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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_forward
- Consider the ring R = {r, s,t) whose addition and multiplications tables are given below. rr t. Then t.s =arrow_forwardThe number of zero divisors of the ring Z4 + Z5 isarrow_forwardLet R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a zero-divisor. We start the proof by (ab) e = 0, e# 0. Which of the following is a true statement in the proof? If ac = 0 then a = 0 If ac + 0 then b = 0 If ac = 0 then a is a zero divisor If bc = 0 then a = 0arrow_forward
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