Q1) Let D = M₂ (Z2)be a ring find the order of D and all the idempotent elements.
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- [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]An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
- 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.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.)18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
- Exercises If and are two ideals of the ring , prove that is an ideal of .24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .
- a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].19. Find a specific example of two elements and in a ring such that and .21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.