(1) For every ring R and R-module M below, determine whether M 0 and prove your answer. (a) R= Z, M = Z/(13) ®z Z/(21);
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- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .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 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+y422. Let be a ring with finite number of elements. Show that the characteristic of divides .15. Let and be elements of a ring. Prove that the equation has a unique solution.
- Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .. 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 .[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]
- 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 ].21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.Prove that if R is a field, then R has no nontrivial ideals.