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- 15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
- Prove that if R is a field, then R has no nontrivial ideals.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 .)An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.