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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]22. Let be a ring with finite number of elements. Show that the characteristic of divides .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] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]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]19. Find a specific example of two elements and in a ring such that and .
- 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.)46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.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 .)
- Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.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.32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .