Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 13, Problem 7E
Let R be a finite commutative ring with unity. Prove that every nonzeroelement of R is either a zero-divisor or a unit. What happens ifwe drop the “finite” condition on R?
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Question 1. Solve the system
-
x1 x2 + 3x3 + 2x4
-x1 + x22x3 + x4
2x12x2+7x3+7x4
Question 2. Consider the system
= 1
=-2
= 1
3x1 - x2 + ax3
= 1
x1 + 3x2 + 2x3
x12x2+2x3
= -b
= 4
1 For what values of a, b will the system be inconsistent?
2 For what values of a, b will the system have only one solution?
For what values of a, b will the saystem have infinitely many solutions?
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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