Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])
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- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .Show that the ideal is a maximal ideal of .
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- Exercises If and are two ideals of the ring , prove that is an ideal of .[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]8. Prove that the characteristic of a field is either 0 or a prime.
- 14. Let be an ideal in a ring with unity . Prove that if then .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 .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.)