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

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

Q: Prove that subring R[x2, x3] not isomorphic to R[x].

A:

Q: Determine whether T={(a, - a) | aEZ} is a subring of ZxZ.

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Q: is a maximal ideal of Q[z].

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

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Q: z6  a subring of z12??

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Q: Determine the smallest subring of Q that contains ½. (That is find the subring of S with the…

A: We have to determine the smallest subring of Q that contains 12. If T is any subring containing 12…

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

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Q: аа Let R = a, bEZ}. Prove or disprove that R is a subring of M,(Z).

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Q: Prove that Q is not isomorphic to Z.

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Q: c) The operation (ao + a1r + azx², bo + bịx + b,x²) = aobo + a¡b1 – azbz defined on P2 has all the…

A: We have to check whether the operation defined on P2 satisfies all the properties to be an inner…

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

A: 3

Q: Find a bosis for the nullspace of the morkit 5-23 (-12 -2-10 4 -6J

A:

Q: Prove that no element of ℤ/n is both a zero divisor and a unit.

A: To Determine :-  Prove that no element of ℤn is both a zero divisor and a unit.

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

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Q: is a maximal ideal of Qlz).

A:

Q: The number of zero divisors of the ring Z4 O Z3 is

A: We have to find the number of zero divisors in the ring Z4⊕Z3.

Q: 16. If R is a field, show that the only two ideals of R are {0} and R itself.

A:

Q: Find all solutions of x2 + x = 0 in the ring Z12

A: Given x2+x=0

Q: The set H= {0,2,3}is a subring of (Z6, +6,.6) integer T F ring module 6.

A:

Q: Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and multiplication…

A: (These are two different questions, only required to do the first question. Please post the second…

Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: show that an ideal A is maximal iff A is

A:

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

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Q: 2 5= {(0, 0₂, 93) | DER₁, A₂ER₂, az ER₂ Z Show that sis subring of RxP₂Rz

A: We have to show that S is a subring of ring R1×R2×R3.

Q: is a maximal ideal of Q[x).

A:

Q: Which ring is fx=2x^2-1 irreducible? Z[x] Q[x] Z2[x] Z3[x

A:

Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: 2. Let S= {a + bi | a, b e Z, b is even}. Show that S is a subring of Z[i], but not an ideal of…

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Q: Find the charactes ist c of the ring Z2 Zg

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Q: Is the ring Z/6Z and Z6 are isomorphic.

A:

Q: Show that the power set of 2, 2º, is algebra on 2. an

A: A σ-algebra is a type of algebra so, every sigma algebra is an algebra.        (1)

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

A:

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

A: Given data:- Explain that 2Z ∪ 3Z is not a subring of Z.

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: 35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and…

A: The objective is to show that the first ring of the following is not isomorphic to the second:

Q: Prove the following statement: If x € Z, shares a common nontrivial factor with n, then a does not…

A:

Q: Determine whether T={(a, - a) | aEZ} is a subring of zxZ

A: If (R,+,.) be a ring and S be a nonempty subset of R then S is a subring of R if and only if 1) a∈S…

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

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

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Q: Is Z6 a subring of Z12?

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Q: Consider the ring of polynomia! Q [ Z] 2 -1 €Q[x] is aprincipl ideal 5

A: In the question it is asked to find whether <x2-1> is a principal ideal.

Q: For the ring Z[√d] = {a + b√d | a, b ∈Z}, where d ≠ 1 and d isnot divisible by the square of a…

A:

Q: Show that for any positive integer n, gl,(C) can be written as a direct sum of two ideals, one of…

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Q: What is the field of fractions of Z[x], the ring of polynomials with integer coefficients?

A: Please check the answer in next step

Q: Show that the rings (3Z/60Z)/(12Z/60Z) and 3Z/12Z are isomorphic. Then show tha both isomorphic to…

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Q: 12. Show that the set of all constant polynomials in Z[r] is a subring but not an ideal of Z[x]

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Q: Find all maximal ideals in: (a) Z10 (b) Z21

A: (a) Maximal ideal of Z10= (i) {0,2,4,6,8}   (ii) {0,5}