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- 23. Find all distinct principal ideals of for the given value of . a. b. c. d. e. f.. 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 .31. Prove statement of Theorem : for all integers and .
- Prove that if R is a field, then R has no nontrivial ideals.In the ring of integers, prove that every subring is an ideal.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 .)
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