a. Give an example of a ring
b. Give an example of a noncommutative ring
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Elements Of Modern Algebra
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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
- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4arrow_forward15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .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
- 33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)arrow_forwardProve that a finite ring R with unity and no zero divisors is a division ring.arrow_forward40. Let be idempotent in a ring with unity. Prove is also idempotent.arrow_forward
- Let 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_forwardExamples 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_forward[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,