Enter the multiplicative inverse of the element [10] x + [13] in the ring z-[x]
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- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .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 .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.
- 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.37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.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][Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]Prove that a finite ring R with unity and no zero divisors is a division ring.
- 32. Consider the set . a. Construct addition and multiplication tables for, using the operations as defined in . b. Observe that is a commutative ring with unity, and compare this unity with the unity in . c. Is a subring of ? If not, give a reason. d. Does have zero divisors? e. Which elements of have multiplicative inverses?Write out the addition and multiplication tables for 4.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.)