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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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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- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forward[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]arrow_forwardLet R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)arrow_forward
- 12. Let be a commutative ring with unity. If prove that is an ideal of.arrow_forwardProve that a finite ring R with unity and no zero divisors is a division ring.arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwarda. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].arrow_forward14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.arrow_forward
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