In Exercises 7 through 13. decide whether the indicated operations of addition and multiplication are defined(closed) on the set, and give a ring structure. Il a ring is not formed, tell why this is the case. If a ring is formed,state whether the ring is commutative, whether it has unity, and whether it is a field.10. 22 x Z with addition and multiplication by components

Definition of Ring : If + and x are two binary operators on a non empty set R and ( R, + , x )…

Answered: n Exercises 7 through 13. decide…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 33E: Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition...

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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]
52. (See Exercise 51.) a. Write out the elements of and construct addition and multiplication tables for this ring. (Suggestion: Write for, for in.) b. Is a commutative ring? c. Identify the unity elements, if one exists. d. Find all units, if any exist. e. Find all zero divisors, if any exist. f. Find all idempotent elements, if any exist. g. Find all nilpotent elements, if any exist. Exercise 51. 51. Let and be arbitrary rings. In the Cartesian product of and, define if and only if and , , . Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of and and is denoted by. Prove that is commutative if both and are commutative. Prove has a unity element if both and have unity elements. Given as example of rings and such that does not have a unity element.
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.
39. (See Exercise 38.) Show that the set of all idempotent elements of a commutative ring is closed under multiplication. Exercise 38. An element in a ring is called idempotent if . Find two different idempotent elements in .
[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]
Prove that a finite ring R with unity and no zero divisors is a division ring.
40. Let be idempotent in a ring with unity. Prove is also idempotent.
An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.
Prove that every ideal of n is a principal ideal. (Hint: See corollary 3.27.)
21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition and multiplication as defined in 18, consider the following questions. Is S a ring? If not, give a reason. Is S a commutative ring with unity? If a unity exists, compare the unity in S with the unity in 18. Is S a subring of 18? If not, give a reason. Does S have zero divisors? Which elements of S have multiplicative inverses?
22. Let be a ring with finite number of elements. Show that the characteristic of divides .

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