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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 6.2, Problem 4E
Textbook Problem
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Prove that R contains an idempotent element if and only if R '

To determine

To prove: R contains an idempotent element if and only if R' contains an idempotent element.

Explanation of Solution

Given information:

R and R' are isomorphic arbitrary rings.

Formula used:

i) Idempotent element:

An element x in a ring is called idempotent if x2=x.

ii) Isomorphism:

Let θ be a homomorphism from the ring R to the ring R'. If θ is a one-to-one correspondence (both onto and one-one), then θ is an isomorphism.

Proof:

Let R and R' be isomorphic arbitrary rings.

Therefore, there exists an isomorphism θ:RR' such that

i) θ is a homomorphism from R to R' ;

ii) θ is a one-to-one correspondence (both onto and one-one).

To prove: a) R contains an idempotent element b) R' contains an idempotent element.

a) b): Suppose R contains an idempotent element.

Let a be the idempotent element in R.

a2=a

Now, let θ(a)R' be any element with aR.

θ(a2)=θ(a)

Since θ(a2)=θ(aa) and θ preserve multiplication.

θ(a2)=θ(aa)=θ(a)θ(a)=(θ(a))2

(θ(a))2=θ(a), for aR

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Elements Of Modern Algebra
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Ch. 6.1 - Prove or disprove each of the following...Ch. 6.1 - Exercises If and are two ideals of the ring ,...Ch. 6.1 - Exercises If , is an arbitrary collection of...Ch. 6.1 - Exercises Find two ideals and of the ring such...Ch. 6.1 - Exercises Let be an ideal of a ring , and let be...Ch. 6.1 - Exercises If and are two ideals of the ring ,...Ch. 6.1 - Find the principal ideal (z) of Z such that each...Ch. 6.1 - Let I1 and I2 be ideals of the ring R. Prove that...Ch. 6.1 - Find a principal ideal (z) of such that each of...Ch. 6.1 - 12. Let be a commutative ring with unity. If...Ch. 6.1 - 13. Verify each of the following statements...Ch. 6.1 - 14. Let be an ideal in a ring with unity . Prove...Ch. 6.1 - Let I be an ideal in a ring R with unity. Prove...Ch. 6.1 - Prove that if R is a field, then R has no...Ch. 6.1 - In the ring of integers, prove that every subring...Ch. 6.1 - Let a0 in the ring of integers . Find b such that...Ch. 6.1 - 19. Let and be nonzero integers. Prove that if and...Ch. 6.1 - 20. If and are nonzero integers and is the least...Ch. 6.1 - Prove that every ideal of n is a principal ideal....Ch. 6.1 - 22. Let . Prove . Ch. 6.1 - 23. Find all distinct principal ideals of for the...Ch. 6.1 - 24. If is a commutative ring and is a fixed...Ch. 6.1 - Given that the set S={[xy0z]|x,y,z} is a ring with...Ch. 6.1 - 26. Show that the set of all 2 X 2 matrices...Ch. 6.1 - With S as in Exercise 25, decide whether or not...Ch. 6.1 - 28. a. Show that the set is a ring with respect to...Ch. 6.1 - 29. Let be the set of Gaussian integers . Let . ...Ch. 6.1 - a. For a fixed element a of a commutative ring R,...Ch. 6.1 - Let R be a commutative ring that does not have a...Ch. 6.1 - 32. a. Let be an ideal of the commutative ring ...Ch. 6.1 - 33. An element of a ring is called nilpotent if...Ch. 6.1 - 34. If is an ideal of prove that the set is an...Ch. 6.1 - Let R be a commutative ring with unity whose only...Ch. 6.1 - 36. Suppose that is a commutative ring with unity...Ch. 6.2 - True or false Label each of the following...Ch. 6.2 - True or false Label each of the following...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Each of the following rules determines a mapping...Ch. 6.2 - 2. Prove that is commutative if and only if is...Ch. 6.2 - 3. Prove that has a unity if and only if has a...Ch. 6.2 - Prove that R contains an idempotent element if and...Ch. 6.2 - 5. Prove that contain a zero divisor if and only...Ch. 6.2 - 6. (See Exercise 3.) Suppose that is an...Ch. 6.2 - Assume that the set S={[xy0z]|x,y,z} is a ring...Ch. 6.2 - Assume that the set R={[x0y0]|x,y} is a ring with...Ch. 6.2 - 9. For any let denote in and let denote in . a....Ch. 6.2 - Let :312 be defined by ([x]3)=4[x]12 using the...Ch. 6.2 - 11. Show that defined by is not a homomorphism. Ch. 6.2 - 12. Consider the mapping defined by . Decide...Ch. 6.2 - 13. Consider the mapping defined by Decide...Ch. 6.2 - 14. Let be a ring with unity . Verify that the...Ch. 6.2 - In the field of a complex numbers, show that the...Ch. 6.2 - (See Example 3 of section 5.1.) Let S denote the...Ch. 6.2 - Define :2()2(2) by ([abcd])=[[a][b][c][d]]. Prove...Ch. 6.2 - Let :2() where 2() is the ring of 22 matrices over...Ch. 6.2 - Assume that and are rings with respect to...Ch. 6.2 - Show that the ring S={[abba]|a,b}2() and the ring...Ch. 6.2 - 21. Let be the subring of that consists of all...Ch. 6.2 - 22. Suppose and are rings with unity elements and...Ch. 6.2 - 23. Let be rings and and be a homomorphism. Prove...Ch. 6.2 - 24. Suppose is a homomorphism from to . Let ....Ch. 6.2 - 25. Figure 6.3 gives addition and multiplication...Ch. 6.2 - Figure 6.4 gives addition and multiplication...Ch. 6.2 - 27. For each given value of find all homomorphic...Ch. 6.2 - 28. Suppose is a field and is an epimorphism from...Ch. 6.2 - 29. Assume that is an epimorphism from to ....Ch. 6.2 - 30. In the ring of integers, let new operations of...Ch. 6.2 - 31. Let and be ideals of the ring. Prove that is...Ch. 6.3 - True or False Label each of the following...Ch. 6.3 - True or False Label each of the following...Ch. 6.3 - True or False Label each of the following...Ch. 6.3 - True or False Label each of the following...Ch. 6.3 - True or False Label each of the following...Ch. 6.3 - Find the characteristic of each of the following...Ch. 6.3 - Find the characteristic of the following rings. 22...Ch. 6.3 - 3. Let be an integral domain with positive...Ch. 6.3 - Show by example that the statement in Exercise 3...Ch. 6.3 - 5. Let be a ring with unity of characteristic ....Ch. 6.3 - Suppose that R and S are rings with positive...Ch. 6.3 - 7. Prove that if both and in Exercise 6 are...Ch. 6.3 - 8. Prove that the characteristic of a field is...Ch. 6.3 - Let D be an integral domain with four elements,...Ch. 6.3 - Let R be a commutative ring with characteristic 2....Ch. 6.3 - 11. a. Give an example of a ring of...Ch. 6.3 - 12. Let be a commutative ring with prime...Ch. 6.3 - Prove that n has a nonzero element whose additive...Ch. 6.3 - Let R be a ring with more than one element that...Ch. 6.3 - 15. In a commutative ring of characteristic 2,...Ch. 6.3 - A Boolean ring is a ring in which all elements x...Ch. 6.3 - 17. Suppose is a ring with positive...Ch. 6.3 - If F is a field with positive characteristic p,...Ch. 6.3 - 19. If is a positive prime integer, prove that any...Ch. 6.3 - Let I be the set of all elements of a ring R that...Ch. 6.3 - 21. Prove that if a ring has a finite number of...Ch. 6.3 - 22. Let be a ring with finite number of...Ch. 6.3 - 23. As in the proof of Theorem 6.20, let. Prove...Ch. 6.3 - With S as in Exercise 23, prove that the right...Ch. 6.3 - With S as an in Exercise 23, prove that the set...Ch. 6.3 - Prove that every ordered integral domain has...Ch. 6.4 - Label each of the following statements as either...Ch. 6.4 - True or False Label each of the following...Ch. 6.4 - According to part a of Example 3 in Section 5.1,...Ch. 6.4 - Let R be as in Exercise 1, and show that the...Ch. 6.4 - Show that the I=(6) is a maximal ideal of E.Ch. 6.4 - Show that the ideal is a maximal ideal of . Ch. 6.4 - Let R and I be as in Exercise 1, and write out the...Ch. 6.4 - Let R and I be as in Exercise 2, and write out the...Ch. 6.4 - With I as in Exercise 3, write out the distinct...Ch. 6.4 - Withas in Exercise,write out the distinct elements...Ch. 6.4 - Find all maximal ideals of . Ch. 6.4 - Find all maximal ideals of 18.Ch. 6.4 - Let be the ring of Gaussian integers. Let ...Ch. 6.4 - Let R bethe ring of Gaussian integersas an...Ch. 6.4 - An ideal of a commutative ring is a prime ideal...Ch. 6.4 - Prove that for , an ideal of is a prime ideal if...Ch. 6.4 - Show that the ideal in Exercise 1 is not a prime...Ch. 6.4 - Show that the ideal of is not a prime ideal...Ch. 6.4 - 17. Show that the ideal in Exercise is a prime...Ch. 6.4 - Show that the ideal I in Exercise 2 is a prime...Ch. 6.4 - 19. Show that the ideal is a prime ideal of. Ch. 6.4 - 20. Show that the ideal is a prime ideal of. Ch. 6.4 - Find all prime ideals of . Ch. 6.4 - Find all prime ideals of . Ch. 6.4 - Give an example of two prime ideals such that...Ch. 6.4 - 24. Show that is a maximal ideal of. Ch. 6.4 - Show that is a prime ideal of but not a maximal...Ch. 6.4 - . a. Let, and . Show that and are only ideals...Ch. 6.4 - 27. If is a commutative ring with unity, prove...Ch. 6.4 - If R is a finite commutative ring with unity,...

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