3. In the ring (Z,2, +12,-12 ), the following ideal is maximal ........ (а) (2) (b) (3) (c) (a) and (b) (d) (a) or (b)

Q: 4. The ring (Z, +,.) the ideal (12) containing in the following maximal ideal ... (a) (4) (b) (5)…

A:

Q: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is a…

Q: 17. Find a nontrivial proper ideal of Z x Z that is not prime.

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Q: (5) The ring (Z6, +6,•6 ) has that rad Z, (a) ((0), +6.-6 )

A: Given: To explain the correct option of the given statement as follows,

Q: Prove that the ideal <x2 + 2> is prime in Z[x] but not maximal in Z[x].

A: The prime ideal <x2+2> is not maximal in Z[x] if Z[x]/<x2+2> is isomorphic to the…

Q: (3) The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +,.) (b) ((8), +,.) (c) ((12),…

A:

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +, .) (c) ((0),…

A: The given question is solved with explanation below.

Q: 1) The r ing of in tegers (Z) is not ideal of Yeal umber (R) n O True Fulse

A:

Q: ) Let Z3[i] = {a+ bi | a, b E Z3}. Show that Za[i] is isomorphic to Zs[a]/ (a² + 1). ) Is the ideal…

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Q: Show that N= (3x, y)|x, y ∈ Z  is a maximal ideal ofZ×Z. Note that you must show that it is indeed…

A:

Q: (c) Show that the ideal generated by x² + y² + z² € C[x, y, z] is a prime ideal.

A: If <x2+y2+z2>is not a prime ideal, there should exist g,h∈ℝx, y, z s.t. x2+y2+z2 | gh,…

Q: (7) In the ring (Z, +, .), we get n {P:P non trival prime ideal in Z}: (a) o (b) (Z, +, .) (c)…

A:

Q: (7) In the ring (Z, +,.), we get n {P: P non trival prime ideal in Z} %3D ....... (a) o (b) (Z, +,.)…

A:

Q: What are the maximal ideals in Z12? , O , O , O

A: Solution: Definition: A maximal ideal of a ring R is an ideal M≠R such that there is no proper ideal…

Q: (1) The ring ({0,2,4,..,22}, +24,-24 ) has the following maximal ideal .. (a) ((6), +24,-24) (b)…

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Q: a Prove that I = { : a,b,c,d are even integers} is an ideal of M2(Z). d

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Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: One of the following is an ideal of z1: (0,3} (0,2,4,6} None (9,3,6,9}

A: From given statement

Q: 2. Prove that if AB = AC, then B need not equal C unless A is invertible.

A:

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: 4. Prove that I[a] = {ao + a1x + ..+ ana" a; = 2k; for k; e Z}, the set of all polynomials with even…

A:

Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

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Q: In the ring Z O Z, I = {(a,0)|a € Z} is: O prime not maximal maximal ideal neither prime nor maximal

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Q: EN is an ideal of R.

A:

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: Show that for any nontrivial ideal I of Z[i], Z[i]/I is finite.

A:

Q: Prove that the ideal (x2 + 1) is prime in Z[x] but not maximal in Z[x].

A: To prove that the ideal <x2+1> is prime in Z[x] but not maximal in Z[x]. Use the definition,…

Q: , then A'(V) is a free K – module of rank

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Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?

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Q: show that Q[x]/(3x⁴+2x³+1) is a field. Here (3x⁴+2x³+1) is the principal ideal generated by a…

A: Given factor ring is ℚx3x4+2x3+1 where 3x4+2x3+1 is the principal ideal generated by the polynomial…

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

A:

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0),…

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

Q: 7) There is no proper non-trivial maximal ideals in (Z2₁, because is a maximal ideal in Z2₁, ,0 )…

A: Solution

Q: 4. Prove C(n, r) = C(n- 1,r- 1) +C(n- 1, r) %3D using a combinatorial proof.

A: We know, C(n,r) = n!/((n-r)!r!)

Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

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Q: The ring Z8[i] has no proper ideals True False

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

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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Q: 4Z is prime ideal of 2Z, but it is not Maximal ideal of Z. To fo

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Q: [a Prove that I= { b. : a,b,c,d are even integers} is an ideal of M2(Z). d_ 1.

A: 1) The main aim is to show that I=acbd|a,b,c,d are even integersis an ideal in M2Z. to show that the…

Q: - Let R=(Z9,+9,9). Find 1. Char(R) 2. Nilpotent ideals of R 3. Prime ideals of R.

A: Solution

Q: (3) The ring (4Z, +,.) has the following prime ideal.. (a) ((0), +, .) (b) ((8), +,.) (c) ((12),…

A:

Q: In Z{x}, Let I = {f(x) ∈ Z[x] / f(0) is an even integer} Prove that I = is I prime ideal of Z{x}?…

A: It is given that,  ℤx is a ring and I = f(x)∈ℤx : f(0) is an even integer. (i) Prove that I = x, 2…

Q: Show that 2Z U 3Z is not a subring of Z.

A: show that 2Z∪ 3Z is not a subrings of Z:   Solution:-             Let 2∈ 2Z∪3Z…

Q: The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +, .) (b) ((8),+,.) (c) ((12), +,.)…

A: Prime ideal sometimes behaves like a prime numbers . Let's firstly define prime ideal.

Q: In the ring Z Z , I = {(0, b)|b € Z} is maximal ideal prime not maximal neither prime nor maximal

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Q: let I be a non-Reso ideal.

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Q: What is the principal ideal (3) of z8 = {0,1,2,3,4,5,6,7}? Is it not just z8?

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Q: The set H={0,1,2}is a subring of (Z4, +4,.4) integer ring module 4. T OF O O

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Q: In the ring Z4 O Z4,1 = {(0,b)|b € Z4} is: O neither prime nor maximal O prime not maximal maximal…

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Q: (7) In the ring (Z,+,.), we get n{P:P non trival prime ideal in Z} = ...... (a) o (b) (Z, +, .) (c)…

A: