Question 6. (a) Find all the ideals of a field . Show your working. (b) Prove that if I is an ideal of ring R, then R/I is commutative iff ab-ba I Va, b € R. (c) Given two polynomials f(x) = 2x²-3 and g(x)= x+1 in the polynomial ring Ze[r], find unique polynomials u(x), v(x) € Z[r] such that ged(f(x), g(x)) = f(x)u(x) + g(x)v(x).

Q: 1. Let f (x) and g (x) be two non-zero polynomials in R [x], R being any ring. (i) If f (x) +g (x)#…

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Q: 1. Let f(x) and g (x) be two non-zero polynomials in R[x], R being any ring. (i) If f (x) +g (x)# 0,…

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Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

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Q: (3) (D) Let R be the ring of all functions from R to R. Let I be the subset consisting of those…

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Q: Suppose that for polynomials J1/ 2J 3 over a field F the polynomial f is irreducible. Which is true…

A: Option {A} is correct. If f is irreducible  Than Statement ( A ) is true. Statement 3 and 4 not…

Q: Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations…

A: Consider the quotient ring R1=Z3x/x2+1  and  R2=Z3i=a+bi/a,b∈Z3 with operations…

Q: 4. Determine whether or not each of the following factor ring is a field. (a) Q[x]/{x² – 5x +6) (b)…

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Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

Q: 4. The following polynomial is irreducible in Q[x]: m(x) = x² - 2x²-1 € Q[x] Let's consider the…

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Q: If R is a ring and f(x) and g(x) are of degrees 3 and 4 then f(x)g(x) must have degree 7.

A: Given that R is a ring. deg{f(x)}=3 deg{g(x)}=4

Q: 4. Let f: RS be a ring homomorphsim, I an ideal in R, and J an ideal in S. (i) Prove that f-¹(J) is…

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Q: Let R be a ring with 1 0. Prove or disprove: (a) if R has no ideals other than {0} and R, then R is…

A: Given statement is false. Justification is in step 2

Q: Let f (x) and g (x) be two non-zero polynomials in R [x], R being any ring. (i) If f (x) + g (x) #…

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Q: Can you me please Give an example of a ring R and polynomials f (x) and g(x) in R[x] such that…

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Q: 1. Find the greatest common divisor of p(x) = x³ + 6x2 + 6x +5 and g(x) = 23x²-x-2. Does your answer…

A: To find the gcd of p(x) and q(x), we can use the Euclidean algorithm: First, p(x) /q(x) and  we get:…

Q: (x), J = (x + 1) en K := (x²020 + x) in the polynomial ring R[x]. Rewrite the ideal I · (I + (J^ K))…

A: Aim: To define the principal ideals and in the polynomial ring . And to define as principal…

Q: (a) Let R be a ring. Give the definition of a polynomial in x with coefficients in R. (b) Define the…

A: Answers to all the questions will be done using basic algebraic techniques

Q: 4. Consider P = P(3, V15) = {a + b/15|a, be Z, 3|a} and Z[V15] = {a + bv15|a, b e Z}. (a) Prove that…

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Q: if the following ideals are Maximal, Prime but Prime. You do not have to provide reasons i. (the…

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Q: Determine all the values of the parameter b € R, for which the polynomial f = x³y + bxy³2² belongs…

A: Given information:f(x,y,z)=x3y+bxy3z2Ideal: (x2+2y2,xz−y)To find: The suitable value of the…

Q: For a field F find an irreducible polynomial f(x) which generates the same ideal in F[x] as the two…

A: Polynomial which is divisor of each polynomial.

Q: (i) Find gcd(2,3+5i) in Z[i], (ii) Show that 3 + 4i and 4-3i are associates in Z[i] (iii) Determine…

A: (i).

Q: Question 6. (a) Find all the ideals of a field . Show your working. (b) Prove that if I is an ideal…

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Q: Let an ideal I = {(a, 0): a ∈ Z} of the ring R = Z ⊕ Z be given. I, R is a prime Is it the ideal

A: To prove: I is an ideal and it is prime ideal of R. Here, R=Z⊕Z Let ,I=a,0:a∈ℤ A non empty subset I…

Q: Consider the quotient ring R = Q[x] x² − 4 > (a) Give an example of a zero divisor in R. Explain why…

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Q: Question 2. (I) Let R be a ring. Show that every unit is not a zero divisor.

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Q: (2) (A) Let R be a ring with the property that r² = r for all r € R. Show that (i) if a € R, then a…

A: A ring is a triplet that is R,+,· where R is a set and uses two operations. It follows the following…

Q: SOLVE STEP BY STEP IN DIGITAL FORMAT 6. A small metal bar, whose initial temperature was 20°C, is…

A: Given,Initial temperature is .Then,Then, by the formula of of Newton's law of heating we…

Q: 3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[T]/I provides an…

A: Using properties of Ring homomorphism we solve this problem.

Q: (6) The ring (Z,2, +12, 12) is .. ..... (а) field (b) semi-simple (c) simple (d) No Choice

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Q: Is the idcal (x² + 1, x + 3) C Z[x] a principal idcal? Explain. The ring Z[x]/(x² +1, x+3) is…

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Q: (4). (5). mial ring C[x]. (6). The ring F[x]/(x²) has a unique maximal ideal. The set {f € C[x] |…

A: Asper the question we are given the following statements which we have to check for true/false with…

Q: Determine whether the given polynomial quotient ring R is a field or not. If R is a field, provide a…

A: We have used some properties and implications .