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- 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 .[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
- An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.2. Prove that is commutative if and only if is commutative.