• Determine the elements of [i]/(1+2i). Note: (1 + 2i) is Į [i] is a Hint: It has an I deal . Gaussian Ring. 5 elements.
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- 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.Find the characteristic of each of the following ring: a. b. c. M2() d. M2() e. M2(2) f. M2(3)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 ].
- 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[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]15. Let and be elements of a ring. Prove that the equation has a unique solution.
- 23. Find all distinct principal ideals of for the given value of . a. b. c. d. e. f.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 .)Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]
- Exercises If and are two ideals of the ring , prove that is an ideal of .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.)An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.