In the ring Z[x]/: 1- The inverse of x is:

Q: Let I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.

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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: Explain why the polynomial rings R[r] and C[x] are not isomorphic.

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Q: In the ring Z:[x] 1- The inverse of xis: a) x+1 b) x+1 c) x d) x 2- The value thatreplacesx is a)…

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Q: In the ring Z[x]/ 1- The inverse of x is: a) x+1 b) x2 +1 c) x d) x? 2- The value that replaces x is…

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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: Q2) Let (M₂ (R), +..) be a ring. Prove H = {(a) la, b, c €. = {(ab)la, b, c € R}is a subring of (M₂…

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Let f(x)= x³ + 2x+ 1 and g(x)= 4x + 1 be two polynomials in a ring (Zs[x],+ ,.). Find q(x) and r(x).

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Q: The ring Zs(i) has no proper ideals True False

A: Given statement is false. Justification is given in step two.

Q: The ring Z is isomorphic to the ring 3Z

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Q: In the ring Z;[x]': 1- The inverse of x is:

A: Answer :

Q: やthe ring KLx,り,z3 wherc K is a field. Prove (x2 - (y1)) that is a K[x, り,そI . Prime ideal of

A: Given:- Ring  K[x ,y ,z]   where k is a field. To Prove:-  [x z-(y2+1)]

Q: In Z[x], the ring of polynomials with integer coefficients, let I = {f(x) E Z[x] I f(0) = 0}. Prove…

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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: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

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Q: Q1/ Let (Maxz (IR), +) be a ring; Is it Integral domain ?

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: In the ring Z[x] 1- The inverse of x is: a) x+1 b) x+1 c) x d) x 2- The value that replacesx is a)…

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

Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

Q: The ring Z is isomorphic to the ring 3Z O True False

A: Solution:

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: Let R be a ring with 1. Show that R[z]/ (x) ~ R.

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Q: The ring 5Z is isomorphic to the ring 6Z

A: Since the z is same for both and the multiplier is different i.e. 5 &amp;6.

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: The number of nilpotent elements in the ring Z23 is

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

Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

Q: The ring Zs[i] has no proper ideals

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Q: Let R be ring, then R is T F imbedded in the polynomial ring R[X].

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Q: Give the splitting field of x² – 1 over Z2, Zz and Z5. -

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Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: Is the mapping from Z5 to Z30 given by x --&gt; 6x a ring homomorphism?Note that the image of the…

A: Given a mapping,

Q: Explain why the polynomial rings R[x] and C[x] are not isomorphic.

A: Suppose there exists an isomorphism φ:R[x]→C[x]. Because isomorphisms are by definition surjective,…

Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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Q: The ring Zs[i] has no proper ideals True O False O O

A: The given statement is ,The ring Z8i has no proper ideals.

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: In the ring Z[x]/ 1- The inverse of x² is a) x+1 b) x² +1 с) х d) x2 2- The value that replaces x is…

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Q: For which value of k is the quotient ring Z5[x]/ (x^3 + 2x^2 +kx +3) a field?

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Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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Q: Is the mapping from Z5 to Z30 given by x → 6x a ring homomorphism? Note that the image of the unity…

A: In the given question we have to tell that " Is f(x)=6x is a ring homomorphism from (Z5,⊕5,⊗5) to…

Q: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…

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

A: Given: R is commutative ring

Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: The number of nilpotent elements in the ring Z23 is:

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Q: In the ring Zg [x] . Show that 1+2x is a unit.

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Q: The ring Zs[i] has no proper ideals True False O O

A: We check whether Z8[I] has proper ideal.

Q: In the ring Z[x] +1> 1- The inverse of x² is a) x+1 b) х +1 с) х d) x2 2- The value that replaces x…

A: ℤ2 [ x ]  is the ring of polynomials with coefficients from ℤ2 = { 0 , 1 }  and &lt; x3 + 1 &gt;…

Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…

Q: What is the field of fractions of Z[x], the ring of polynomials with integer coefficients?

A: Please check the answer in next step

Q: The ring Z-[i] has no proper ideals

A: