(1) Let I be a proper ideal of the commutative ring R with identity. Then I is a if and only if the quotient ring R/I is a field. ***** (i) prime ideal (ii) primary ideal (iii) Maximal ideal (2) Z/(5) =........ (i) (5) (ii) 5Z (iii) {[0],[1], [2],[3],[4]} (3) If R is an integral domain has non zero characteristic, then Char(R)=..... (i) 5 (ii) 4 (iii)9

Q: (e) Let G = S4 and H = {e, (1 2)}. What is the size of NG(H), the normalizer of H in G? (f) What is…

A: 1. From definition we know that, Normalizer of e= {g∈S4​:geg−1∈H}. Now geg−1=e[∀g∈S4​] gives us…

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: (4)Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to…

A: Given that, K=Z12 and ideals I and J are such that I=([4])=4Z12={0,4,8} J=([6])=6Z12={0,6} Now,…

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: Let Q[x] denote the ring of polynomials with coefficients in Q (you may assume that Q[x] is a ring).…

A: Given that  be the ring of polynomials with coefficients in .We need to determine, which of the…

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

A: 4)False The statement "The ring F[x]/(x^2) has a unique maximal ideal" is false. Explanation: A…

Q: F Let K/L be ideal of quotient ring R/L. If K is prime ideal and contains L, then K/L is prime…

A: We can solve this using defination of ideals ,prime ideals ,quotient ideals

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: 11. Let f (x) E Z[x] with deg (f(x)) > 1. Let f'(x) be the polynomial in Zp,p € Z, obtained from…

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Q: (1) Let I be a proper ideal of the commutative ring R with identity. Then I is a ........ if and…

A: To choose correct option form given question.

Q: Let f(x) = x³ + x² - 4x − 4, g(x) = x¹ + 3x³ + 5x² +9x + 6 and consider the ideal I = (f(x), g(x))…

A: Let ,  and consider the ideal  generated by f and g in the rings  and .Since these are both…

Q: (5) I is maximal ideal of ring R if and only if R = (a, I) for any..... a in R O a in I a in R-{I} o…

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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: ! (9) Let K be integer ring module 12 and let I-([4]) and J-([6]) be ideals of K. Then ([0]) (i) I +…

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Q: Auume we know that m(X) = X4 + X + 1 ∈ Z2[X] is irreducible. Let K denote the field Z2[X]/(m(X)) and…

A: a) The identity is (b1+b2+...+bk)2=b12+b22+...+bk2(b1​+b2​+...+bk​)2=b12​+b22​+...+bk2​.  b) The…

Q: Problem 2. (a) Verify that the set Q(V3) = {r+sv3 |r,s € Q} is a subfield of the real numbers R.

A: By Bartleby policy I have to solve only first one as these 2 are unrelated and lengthy…

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: (13) Let R be the set of all matrices with rational entries, M₁(Q), of the form abe 0 a b 00 a Show…

A: 1)R is said to be commutative if you can swap them i.e  ab=ba. 2)R/A(Quotient Ring) is a field if…

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

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Q: Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove…

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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: 9. Use the field norm to show: a) 1+/2 is a unit in Z [/2] b) -1+v-3 is a unit in Z [1+-3 ]

A: Use the field norm to show thata1+2 is a unit in 2.b-1+-32 is a unit in -1+-32.

Q: 1. Let F be the field with 2 elements. The number of (ordered) bases of F" over F is (а) п (b) п?…

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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: Let R be a ring such that for each a eR there exists XE R such that a'x = a. Prove the following :…

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Q: ring with identity and I be ideal of R. Then I is prima and only if every invertib R/I is a…

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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: 4. (a) Let S be a commutative ring with 1 and let s E S. Assume that sn = 0 for some integer n ≥ 2.…

A: The objective of this question is to prove that if s is an element of a commutative ring S such that…

Q: Let f (x) = x'+3x+ 2n+ ã and %3D 2 9(x) = x + 2+ 1 E L! [x]. q (x) g(x)+ Y(x) where either os deg…

A: We are given : f(x)=x4+3x3+2x2+2g(x)=x2+2x+1 Now dividing f(x) by g(x), we have  : (Long Division)…