(e) Let G = S4 and H = {e, (1 2)}. What is the size of NG(H), the normalizer of H in G? (f) What is the characteristic of the ring Z/3ZXZ/2Z? (g) Ifo: Z/14ZZ/10Z is a ring homomorphism, then what are the possibilities for |ker()|?

Q: 7. (a) Let be an increasing sequence of ideals in a ring R. Prove that the union of all these 1, is…

A: Both the sub-parts are solved below.

Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

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Q: LetΦ be the ring homomorphism from Z[x] to Z given by Φ(f(x)) =f(1). Find a polynomial g(x) in Z[x]…

A: Given: ϕ is a ring homomorphism ϕ:ℤ[x]→ℤ defined as ϕ(f(x))=f(1). To find: A polynomial g(x) such…

Q: 4. (the Dirichlet ring) Consider the set of all arithmetic functions, R, with the oper- ations of…

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Q: 10. Determine if g(x) divides f(x) in the given rings of polynomials (show your work): a) g(x) x55x4…

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Q: Let R be a ring and define R* = ∆R = {(r, r) : r ∈R }.1 (a) Suppose R = Z6 and list the members of…

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Q: Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations…

A: Consider the quotient ring R1=Z3x/x2+1  and  R2=Z3i=a+bi/a,b∈Z3 with operations…

Q: Can you me please Give an example of a ring R and polynomials f (x) and g(x) in R[x] such that…

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Q: (x), J = (x + 1) en K := (x²020 + x) in the polynomial ring R[x]. Rewrite the ideal I · (I + (J^ K))…

A: Aim: To define the principal ideals and in the polynomial ring . And to define as principal…

Q: Give an example of a ring Zn and polynomials f, g ∈ Zn[x] such that f, g do not satisfy relation…

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Q: 4. Consider P = P(3, V15) = {a + b/15|a, be Z, 3|a} and Z[V15] = {a + bv15|a, b e Z}. (a) Prove that…

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Q: M n of ring (5 Marks) (b) Let (R,+,) and (R', +', '') be any two rings, let f: RR' be a ring…

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Q: Let p: Q[x] → M2(R) be defined by p(ao + a1 + a2x² + ...+ anx") = ( ) ao (1) Write down o(2 +x –…

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Q: Consider the quotient ring R = Q[x] x² − 4 > (a) Give an example of a zero divisor in R. Explain why…

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Q: 3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[T]/I provides an…

A: Using properties of Ring homomorphism we solve this problem.

Q: 4. (a) Let S be a commutative ring with 1 and let s E S. Assume that sn = 0 for some integer n ≥ 2.…

A: The objective of this question is to prove that if s is an element of a commutative ring S such that…

Q: 14. Find all units of the ring C(R) = {ƒ : R → Rf is continuous}, where the operations are the usual…

A: A unit is an element which has multiplicative inverse. Using this definition we find the units in…