Problem 8. Prove that if R is a finite ring with unity then any prime ideal is a maximal ideal.
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- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
- Show that the ideal is a maximal ideal of .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 .). 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 .