Show that in the factor ring Z[x] /(2x+1), the element x+(2x+1) is a unit.

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

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Q: Let R be a ring with unity. Show that (a) = { £ xay : x, y e R }. finite

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Q: Find all values of a in Z5 such that the quotient ring Z5[æ]/(p(x)) where p(x) = x³ + x? +ax + 4 is…

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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 Z is isomorphic to the ring 3Z

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

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

A: Answer :

Q: Give an example of a non-commutative ring R without unity such that (xy)^2 = x^2.y^2 for all x,y in…

A: We consider the example of a non-commutative ring R without unity such that (xy)2 = x2y2 for all x…

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: The ring Zpq²r has exactly-----------maximal ideals 1 2 3

A: 3

Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: 6. Consider the ring of polynomials with rational numbers as coefficients, Q[x]. Set R = {f(x) E…

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Q: 3. Explain why the polynomial rings R[r] and C[r] are not isomorphic.

A: This is a problem of Abstract Algebra.

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: Q15: Define the concept of ideal and then show that the set I = is not an ideal of the ring of all 2…

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

A: Solution:

Q: Let R be a ring with unity. Show that (a) = { £ xay: x, y e R }. finite

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Q: Let R be a ring with unit. Define 2R ≔ 1R + 1R. In most rings, 2R ≠ 1R. Figure out what the…

A: Please check the answer in next step

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

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Q: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…

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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 &6.

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: The ring Z[/2] is an ED with d(a + bv2) = |a² – 21³| . Show that the equations r² – 2y² = 1 and r² –…

A: The objective is to show that equations x2-2y2=1  and  x2-2y2=-1  each have infinitely many integer…

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

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Q: Let a and b be elements of a ring R. Prove that the equation a+ x= b has a unique solution.

A: a and b are elements of ring R. We have to show that (a+x)=b has a unique solution.

Q: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

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: Prove that the numbers 3 and 7 are indecomposable in the ring Z[V-5], but the number 5 is…

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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 commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: The ring 3z is isomophic to the ring 5Z False True

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Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.

A: R1 and R2 are subrings of the ring R, prove that R1∩R2 is a subring of R

Q: Describe all ideals of the following quotient ring: Q[x]/(x² – 5x + 6)

A: Given:Q[x](x2-5x+6)

Q: Q1: Find all tHhe idempotent and nilpotent e le ment of the ring Z5

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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: Q3: Prove that the ring of rational numbers (Q, +,.) is division ring

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

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Q: Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is…

A: Given R be a commutative ring with unit element. If f(x) is a prime ideal of R[x] then we have to…

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

A: The given ring is ℤ2x/x3+1. The objective is to find the inverse of x and to find Uℤ⊕ℤ.

Q: Prove that Q[x]/<x2 - 2> is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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Q: b. Let R be a nontrivial ring such that, for each 0 + a E R there exists unique element x in R such…

A: Note: According to Bartleby guidelines; for more than one question asked, only the first one is to…

Q: 6. Let R Z/8Z and consider the polynomial ring Rc]. Show that the polynomial %3D 1+ 28x is…

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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: The ring 3z is isomorphic to the ring 5z True False

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Q: 3. Let R be any commutative ring with unity, and let T[r] be the subset of all polynomials with zero…

A: Given that R is a commutative ring with unity, and T[x] be a the subset of all polynomials with zero…

Q: Factor the polynomials x* +1 and x° –1 into products of non-decomposable polynomials in the ring…

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!