Examples 5 and 6 of Section 5.1 showed that
Let
a. Prove that multiplication in
b. Is
c. Does
d. Is
e. Is
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Elements Of Modern Algebra
- [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]arrow_forwardExamples 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]arrow_forwardExercises If and are two ideals of the ring , prove that is an ideal of .arrow_forward
- 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.arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward
- 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.)arrow_forward24. 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 .)arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forwarda. 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 ].arrow_forwardExercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .arrow_forward
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