Suppose that R is a commutative ring without zero-divisors. Showthat all the nonzero elements of R have the same additive order.

Q: The image of a commutative ring under homomorphism may be noncommutative False True

A: As you asked multiple questions , I solved the first one. So you asked "the image of a commutative…

Q: 7. Prove that the set of all elements in a ring R that are not zero divisors is closed under…

A: Bb

Q: In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.

A:

Q: The number of zero divisors of the ring Z4+Z4 is?

A:

Q: * (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no…

A: True

Q: Let n be a positive integer. Show that there is a ring isomorphismfrom Z2 to a subring of Z2n if and…

A:

Q: Let a and b be idempotents in a commutative ring. Show that eachof the following is also an…

A:

Q: Let R be a Boolean ring with unity e. Prove that every element of R except 0 and e is a zero…

A: Let R be a Boolean ring with unity e. To prove R is a Boolean r ring with unity e, so every element…

Q: A ring (R. +.) .) is commutative if addition is commutative in R. O True O False

A: Solve the following

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

A:

Q: Decide whether ZxZ = {(n,m) n,me Z} with addition and multiplication by components, give a ring…

A:

Q: If R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.

A:

Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…

A: To prove that every prime ideal of R is maximal.

Q: Give an example of a noncommutative ring that has exactly 16 elements.  Prove that it is a…

A: Ans:   Let R=M2 Z2.This is a finite 16 elements   non commutative ring with identity…

Q: If a is an idempotent in a commutative ring, show that 1 - a is alsoan idempotent.

A: Here given

Q: The number of zero divisors of the ring Z4 O Z3 is

A: We have to find the number of zero divisors in the ring Z4⊕Z3.

Q: Show that the nilpotent elements of a commutative ring form a subring

A: We have to prove that the nilpotent elements of a commutative ring form a subring. In order to do…

Q: (b): Define reducible and irreducible elements in ring R and find all the reducible and irreducible…

A:

Q: 3) Given a commutative ring with unity 1 in R; where R is a ring with two maximal ideals M₁ and Show…

A:

Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: 4. Prove that a zero divisor in a ring cannot be a unit.

A:

Q: Show that in a Boolean ring R, every prime ideal P is not equal to R is maximal.

A:

Q: In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.

A:

Q: Construct the adition and multiplication tables for the ring Z/4Z.

A: Given: A ring ℤ/4ℤ. To construct: Addition and multiplication tables for the ring ℤ/4ℤ.

Q: The set of all zero divisors of the ring Z6 is اختر احدى الاجابات O {2, 3, 4} O (1, 3, 5) O {1, 2,…

A: Z6={0,1,2,3,4,5}Since 2 · 3 ≡6≡ 0 (mod 6) and 3 · 4 ≡12≡ 0(mod 6)However, 1 and 5 are not zero…

Q: Show that a commutative ring with the cancellation property (under multiplication) has no…

A:

Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

Q: Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.

A:

Q: Suppose that a and b belong to a commutative ring and ab is a zero-divisor.Show that either a or b…

A:

Q: (B) Give an example for a commutative ring with identity.

A:

Q: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…

A:

Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

Q: The number of zero divisors of the ring Z4 Zg is 11

A:

Q: If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative…

A:

Q: The number of zero divisors of the ring Z4 Z2 is O 5 O 1

A:

Q: The number of zero divisors of the ring Z4 O Z, is

A:

Q: Show if R is a commutative ring then R[x] is also a commutative ring.

A:

Q: Give an example of a noncommutative ring with unity.

A:

Q: The number of zero divisors in the ring Z100 is:

A: We know that

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A:

Q: determine all ring homomorphisms from Q to Q

A: Determine all ring homomorphisms from Q to Q.

Q: elements

A:

Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

A:

Q: 5- The image of a commutative ring under homomorphism is commutative a) True b) False a) True O b)…

A:

Q: The number of zero divisors of the ring Z4 + Z5 is

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Q: Let R be a commutative ring with more than one element. Provethat if for every nonzero element a of…

A: We have given commutative ring R with more than one element. We have to prove R has a unity and…

Q: 14) Any subring of a commutative ring is commutative. True False

A: I have given an answer in step 2.

Q: Prove that the set of all elements in a ring R that are not zero divisors is closed under…

A: We will prove that the set of all elements in a ring R that are not zero divisors is closed under…

Q: Suppose that R is a commutative ring without zero divisors . show that characterstic of R is 0 or…

A: suppose that R is a commutative ring without zero divisors to show- characteristic of Ring R is 0 or…

Q: Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides…

A: Let a, b, and c be elements of a commutative ring where a is a unit. Suppose that bdivides c. Then c…