Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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- Write 20 as the direct sum of two of its nontrivial subgroups.arrow_forward22. Let be a ring with finite number of elements. Show that the characteristic of divides .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_forward
- Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][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_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forward
- 11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forwardA Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.arrow_forward
- 7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.arrow_forward11. Show that defined by is not a homomorphism.arrow_forward15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .arrow_forward
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