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- Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .Exercises If and are two ideals of the ring , prove that is an ideal of .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 ].
- 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+y418. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
- 19. Find a specific example of two elements and in a ring such that and .[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]Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
- Exercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.Exercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these conditions are satisfied: is nonempty. and imply that and are in .