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

Q: 7. Find all distinct principal ideals of Z14

A: Since Z14 are cyclic group with respect to addition so all ideals of Z14 are principal ideals

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: * find z- transform for -4 x (n) = { 2 1 2 3} x(2) = 2 +z + 2z-2 + 3z-3 x(2) = 222 + z +=+ 2 + 3Z'…

A:

Q: give an example of An ideal in Z that is not a prime ideal.

A: The objective is to write an example of an ideal in ℤ that is not a prime ideal.

Q: Find the QR decomposition and prove -3 -5.

A:

Q: Find

A: We know that,  All ideals of Zn are kZn, Where k is divisor of n.

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: to find the z-transforms of following: {ak-3} (3k-4)

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Q: The ring Z,2, has exactly-- --maximal ideals O 2 3.

A:

Q: Find the splitting field x4 - x2 - 2 over Z3.

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Q: find the generated of zy zs+Z7

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Q: Define principal ideal ring.

A: Ideal: A subring A of a ring R is called a (two-sided) ideal of A if for every r∈R and every a∈A…

Q: Find all maximal and prime ideals of Z10-

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Q: Which of Zs, Z2o are cyclic? 18, '20

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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: is the ring 2Z isomorphic to the ring 4Z??

A: We can apply definition

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: Is cartesian product of prime ideals of a ng also a prime ideal? prove or give a counter example.

A:

Q: The z-transform of Z{3k} O 1/(z-1)^2 O 3z/(z-1)^2 3z/(z-3)^2 3z/(z-1) z/(z-1)^2 O z/(z-1) O…

A: This question is related to Z transform,  we will solve it using given information.

Q: 2) A ving having no proper ideals Called single Ying is D True O Fulse i

A: A ring having no proper ideal is called single ring. to determine whether the statement is true or…

Q: The z-transform of [n] = u[-n] is

A: Using Z transform

Q: Is the polynomial x2 – 4 in the ideal generated by the polynomials x + 22 – 4x – 4, x – 22 – 4x + 4,…

A:

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

A: Theorem for Sebring Test: For any ring R, a subset S of R is a subring if and only if: it is closed…

Q: What is the difference between the notation and {3} .What is the difference between the notation…

A: Here the notation <3> is represent ideal of Z generated by 3      i.e <3> = 3Z is ideal…

Q: Prove that Z[2√-3] is not a principal ideal domain

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Q: Explain why (or prove) {∑nk=3 akxk|ak ∈ R} is not a prime ideal in R[x].

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Q: state and prove the Cauchy's Residue Theorem?

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Q: (i) find gcd(2,3+5i) in Z[i], (ii) show that 3+4i and 4-3i are associates in Z[i] , do all , dont…

A: Solution (i)

Q: The ring Zpg²r has exactly-- ---maximal ideals 2 1 4

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

Q: Theorem 4.8.1 Irrationality of V2 V2 is irrational.

A: For any integer n such that n² is even then n is also even.

Q: Explain why 3Z is not an ideal in Q

A:

Q: It is known that 28= {0, 1, 2, 3, 4, 5, 6.73 is a Ring. H = { 0,43 is a subring of Z8. show that…

A:

Q: Find every maximal ideal of Z, O Z7.

A: Maximal ideal: Let R be a commutative ring and M≠R⊲R then we say M is maximal ideal of R, if…

Q: The ring Z,3 has exactly-------------maximal ideals

A:

Q: See picture

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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: Not prime ideal of R

A:

Q: Find all subfield of Z14

A: Z14 is not a  field reas is given on the next step.

Q: Prove that 8x3 - 6x - 1 is irreducible over Q.

A:

Q: The ring Z,2, has exactly--- --maximal ideals 1 2.

A:

Q: Prove that Z[x] is not a principal ideal domain.

A:

Q: What are all of the distinct generators of Z17 with respect to multiplication? What are all the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Explain the relationship between ideal and subring

A: A subring S of a ring R is a subset of R which is a ring under the same operations as R.   An ideal…

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: 7. Show that the splitting field Q(√2, √3) of (X2-2)(X2-3) is a Galois extension of the rational…

A: If there exist a polynomial of degree 4 generated by this splitting field, then we say this is an…

Q: Prove directly that a maximal ideal is irreducible.

A:

Q: O Principal Ideal Domain

A: Using theorem - Polynomial ring R[x] is pricpal ideal domain iff ring R is field itself Now given D…

Q: Prove that I= (2+ 2i) is not a prime ideal of Z[1].

A: It is given that, I=2+2i is not a prime ideal of ℤi. Definition used: A prime ideal is an ideal I…

Q: The ring Zp2, has exactly-----------maximal ideals

A: