sider the integers Z as a Z-module. Let n be a non-zero integer and define a function fn :ZZ by fn(a) = na (a) Prove that fn is a Z - module homomorphism. (b) Use fn to prove that Z and nZ are isomorphic as Z - modules (c) Let g: Z→ Z be any Z-module homomorphism. Prove that g = fn for some integer n (Hint: consider g(1))

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A: Sol

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A: Suppose z is implicitly given as a function of x and y, according to The equation: y−zx3=1−yz2 The…

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Q: 30. (Remainder Theorem) Let f(x) e F[x] where F is a field, and let a E F. Show that the remainder…

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Q: 11. (Exercise 15.3) Let F be a field and p(x) an irreducible polynomial in F[r]. In this…

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Q: Find the kernels and images of the following homomorphisms. Which of the homomorphisms are…

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Q: Let M and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

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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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Q: Let f : R^3 → R^3 be a homomorphism defined by the following matrix in a certain base B.   What…

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Q: a b Let e: (M2(R), + >→ be defined by e c d = a + d• where a,b,c,d €R (a) Prove that e is a…

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Q: 2. Let R = F[X] and define three different R-module structures on F² (call them V1, V2 and V3), with…

A: Given R= F[X]

Q: Let A = Z+ x Z+. Define a relation R on A as follows: For all (x, y) and (z, w) in A, (r, y)R(z, w)…

A: We will prove the given statement.

Q: d) Prove: If Z = Zo is a zero of order n of flz), then it is zero of order 2n of (f(z))²

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Q: 6. Prove or disprove the statements: (a) The function f: Z × Z → Z defined by f(m,n) = m + n +1 is…

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Q: Let :Z₁₂ →Z, be the homomorphism such that (1) = 2. (a) State the First Isomorphism Theorem. (b)…

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Q: 2. Suppose f is holomorphic in an open set UC C and a € U. Show that for any integer n ≥ 0 there is…

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Q: Let    f(x) = x3 + x + 1    be an irreducible polynomial in Z5[x] Let A be a root of f(x) in an…

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A: We take an example to satisfy this question

Q: A = 30 -5 -6 2 6 2 -12 -9 3 6 0 -10 -9 3 75-1 -50 10 -5 -6 4

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Q: 5. Let K = Q(V2, i). Find all isomorphisms of K onto a subfield Q as extensions of the identity…

A: (5) Given, K=Q2, i Let φ:K→K,  So, φ depends on φ2 and φi Now, minimal polynomial of 2 is x2-2, so,…

Q: (Fundamental theorem an module homomorphism) Let M, M' be an R-module homomorphism. Then Ker (f) is…

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