Show that a unit of a ring divides every element of the ring.

Q: Let R[x] be the set of all polynomials over a ring (R ,+, . ). Then, (R[x] ,+ , .) form ring , which…

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Q: Give an example of a subring of a ring, say A, that is not an ideal of A.

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Q: Let (?,+, ⋅) be a ring with additive identity 0. Prove that for all x∈?, 0⋅x=0 and x⋅ 0 = 0.

A: We know that if (R,+,.) is a ring then (R,+) is an abelian group. And in abelian group, cancellation…

Q: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

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: 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: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

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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 and B are ideals of a ring, show that the product of A and B, AB = {a,b, + a,b, +· · · + a,b,l…

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Q: Show that the set of all rational numbers is only a subring but not an ideal of the ring of real…

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Q: a is a unit in a ring R with unity, then a is not a zero divisor

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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: 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: Show that Ø is a ring homomorphism.

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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: The measure u is monotone on the ring. So that µ(A) < µ(B) if ACB

A: Given that The measure is monotone ob the ring,  So we need to consider the following;

Q: Give an example of a polynomial ring Rx and a polynomial of degree n with more than n zeros over R.

A: A ring R is a set with two binary operations addition and multiplication that satisfies the given…

Q: Show that in the factor ring Z[i] / , the element x + is a unit.

A: Given a factor ring Zx2x+1=a0+a1x,+⋯+2x+1 |a1∈Z  As 2x+1+2x+1=0+2x+1 Since,  2x+1∈2x+1   as aH=H iff…

Q: Given that I = {(a,b) E R | a and b are even numbers} is an ideal of a ring R = Z × Z, construct…

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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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: Prove that a commutative division ring must be an integral domain

A: We have to prove that a commutative division ring must be an integral domain.

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.

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

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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 O Z, is

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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: be a Euclidean ring and asb elements of R- gf a and b Let R are non- ze YO asso ciates , Show that…

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

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

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Q: Let R be the set of all functions. It is known that (R,+,×) is a ring where + and × are both…

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Q: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

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

A: We know that

Q: Determine whether (Z,⊕,∗) is a ring with binary operations defined as (x⊕y)=x+y−7,…

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Q: Let a and b be elements in a ring R. If ab is a zero divisor, prove that either a or b is a zero…

A: Given :          ab is a zero divisor where a and b are elements in ring R. Formula :        Zero…

Q: Show that the factor ring Q[r] (P(x)) has a zero-divisor

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Q: Prove that a finite ring R with unity and no zero divisors is a division ring.

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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: If R be a commutative ring with identity and a ∈ R is idempotent different from 0 or 1. Prove that R…

A: Given that R is a commutative ring  with unity. Also, a∈R  is an idempotent element ⇒a2=a ( a≠0,1)…

Q: Show that the centre of a ring R is a sub- ring of R. And also show that the centre of a division…

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Q: An element x in a ring is called an idempotent if x2 = x. Prove that  the characterstic of R is 0 or…

A: An element x in a ring is called an idempotent if x^2 = x

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: Construct the tables of this ring, write the elements in their most simple form, and determine the…

A: To find the elements of the ring ℤ3xx2+x+1, observe that x2+x+1 is  degree two polynomial, therefore…

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

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Q: The ring Zp2, has exactly-----------maximal ideals

A: