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- 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 .)22. Let be a ring with finite number of elements. Show that the characteristic of divides .11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
- For a commutative ring with unity we may define associates, irreducibles,and primes exactly as we did for integral domains. Withthese definitions, show that both 2 and 3 are prime in Z12 but 2 isirreducible and 3 is not.Let P be a prime ideal and R a commutative ring with identity. If R/P is an integral domain, explain why R/P cannot be the ring with one element. Also explain why P is not equal to R in this scenario.i)Prove that a field has only two ideals. ii)Prove that a commutative ring with unity that has only two ideals is a field.