Find all idempotent elements of the ring M_2(R).
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Find all idempotent elements of the ring M_2(R).
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- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.
- 44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]