{[] a,b € Z}. (Note: It is easy to show that R is a subring of M₂(Z), the ing of 2 x 2 matrices with integer coefficients.) Show that the ideal Z= (G)- is a prime deal in the following. et R= a) Show that the map : R→ Z taking []a-bi b is a st a surjective ring homomorphism. b) Show that ker = I. c) Use the First Isomorphism Theorem to show that I is a prime ideal.

Q: 5. Let F = Q(√2, √3) be the smallest subfield of C that contains √2 and √3. (a) Find a basis for F…

A: Given :  F = ℚ2, 3 be the smallest subfield of ℂ that contains 2 and 3 To find : (a) The basis for F…

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A: We know that M22 is the vector space of all 2×2 matrices with real entries and S2 is the space of…

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Q: -se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT

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Q: Give an example of each of the following, and show that your example satisfies the "highlighted"…

A: For the solution follow the next steps.

Q: 1. Consider the set of upper triangular 2 x 2 matrices, with real entries: b B = {(" "); GAVER} a,…

A: We have to prove that U is an ideal in B.

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Q: Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual…

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Q: 2. Let s-{(소)아 2a S = 26 2a Assume that S is a subring of M2(Q). Prove that S is isomorphic to Q[V2]…

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Q: Q15: Define the concept of ideal and then show that the set I = is not an ideal of the ring of all 2…

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Q: Consider the subset D of M (IR) consisting of the diagonal matrices [9]. a. Prove that D is a…

A: Step 1: Step 2: Step 3: Step 4:

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A: Given that , γ:ℂ→ℂ defined by , γx+iy=x2=y2

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Q: {[: :]we2} b a, b E Z} and let o : R → Z be defined by ø a a Let R = a a а — b. (1) Show that ø is a…

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Q: Every submodule of a noetherian (artinian) module is noetherian (artinian).

A: Every submodule of a noetherian (artinian) module is noetherian (artinian).

Q: Let S = {a+bi: a, b € Z and b is even}. Is S an ideal of Z[i]?

A: For S to be an ideal of Z[i], we must have the following conditions satisfied: 1) a∈S, b∈S ⇒a - b∈S…

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Q: Let a E Z, be a zero of the irreducible polynomial x + x +1 over Z,. Show that the Frobenius…

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Q: = {[0]m [1]m, [m - 1]m} the ring of integers modulo m. Z4 XZ6. Let : RS be the map Denote by Zm…

A: To summarize:The image of (\theta) is the set of all pairs (([x]_4, [y]_6)) where ([y]_6) can be…

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