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

Q: 6. Prove or disprove: the set of all subsets of R is a ring with respect to the operations A…

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

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

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Q: Prove that a division ring has no zero divisors.

A: Answer: Proof:

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

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Q: The number of zero divisors of the ring Z, Z5 is О 1 O 5

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

A: Solve the following

Q: Define reducible and irreducible elements in ring R.

A: Remark: An integral domain is a commutative ring with unity which does not contain any…

Q: An example on a finite non- commutative ring without identity is:

A: Matrices are noncommutative  So we can choose set of matrices  Not if we take Mn[R] then this is non…

Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: Prove or Disprove that is a commutative ring, but without unity, and is not a field

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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: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, (a, b), (c, d)ER^2 then (a, b)*…

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Q: Show that if R is a ring with unity and N is an ideal of R such that N R, then R/N is a ring with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

Q: Give an example to show that the factor ring of a ring with zero divisors may be an integral domain.

A: (i)4ℤ8ℤ≈ℤ2 (ii) ℤ6ℤ

Q: 1. State the Cancellation Axiom for the multiplication. 2. Show, justifying each step, that the…

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Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, V (a, b), (c, d)ER^2 then (a,…

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Q: If A and B are ideals of a commutative ring R, define the sum of A and B as: A+B = {a+b| a € A, b €…

A: Let A and B be two ideals of a ring R. If  A and B are ideals of a commutative ring of R with unity…

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

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Q: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…

A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.

Q: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

Q: Either prove or give a counterexample: If R is a ring with zero divisors, then R[x] is a ring with…

A: (A) Given that R is a ring with zero divisors.

Q: prove or disprove that the smallest non commutative ring is of order 4

A: Using the Result : " All rings of order p2 ( p is any prime) are commutative"   As 4=22 and 2 is a…

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

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

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Q: The ring 5Z is isomorphic to the ring 6Z True O False

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Q: Prove or disprove that Q(√3) and Q(√-3) are ring-isomorphic.

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Q: B. Show that each homomorphism from a field to a ring is either one to one or maps everything nnto 0…

A: 18 Suppose we have a homomorphism φ : F → R where F is a field and R is a ring (for example R itself…

Q: Show that in a ring R cancellation laws hold if and only if R has no zero divisors.

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Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

Q: Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 =…

A: Given Suppose that R is a ring with nonzero-divisors and that R contains a nonzero element b such…

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

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Q: Exercise 2.46.1 Let R be a commutative ring. Suppose that every nonzero element in R is invertible.…

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Q: Provide an example of the following if possible. (a) An infinite ring R such that Char(R) = 97. (b)…

A: Note : for option (c) Take R=Q is simple ring and Q has infinitely many no simple subring…

Q: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

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Q: Let Ebe the set of even integers with ordinary addition. Define a new multiplication on E by the…

A: Let E be the set of even integers where addition is defined as ordinary addition and multiplication…

Q: Give an example of a commutative ring without zero-divisors that is not an integral domain.

A: Integral domain: Let D be a ring. Then D is an integral domain, provided these conditions hold: 1. D…

Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring * O True O…

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Q: Find 5 examples of a ring in which {0R} is maximal

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Q: 1- Let (R,+,-) be aring which has property thut a=a, Ua ER.prove thatR is Commutabive ring (Every…

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

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

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Q: Suppose I,J be ideals of a commutative ring R. Prove that IJ cInJ.

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

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let R be a commutative ring with unity  If I is a prime ideal of R, prove that I[x] is a prime…

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

A: I have given an answer in step 2.

Q: Let N be the set of all nilpotent elements in a commutative ring R. Show that N is an ideal of R.…

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Q: show  that a 2 by2 matrix is a  ring and further is non cummutative

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