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- a. 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 ].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]19. Find a specific example of two elements and in a ring such that and .
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.9. Find all homomorphic images of the octic group.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 .
- 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.15. Let and be elements of a ring. Prove that the equation has a unique solution.22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- Write out the addition and multiplication tables for 5.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?A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.