(6)lf K is a simple ring, then K is a Jacobsen radical ring. T O F
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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 ].Exercises If and are two ideals of the ring , prove that is an ideal of .19. Find a specific example of two elements and in a ring such that and .
- 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+y415. Let and be elements of a ring. Prove that the equation has a unique solution.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.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 .Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .
- 14. Let be an ideal in a ring with unity . Prove that if then .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]Find the characteristic of each of the following ring: a. b. c. M2() d. M2() e. M2(2) f. M2(3)