Let R be a ring and S be set. Denote by RS the collection of all functions f: S→ R. Equipped RS with the pointwise addition and pointwise multiplication f+g: S → R S → f(s) + g(s) R S → f(s)g(s) fg S → and for any f,g: S→ R € RS. Prove that RS is a ring with respect to the operations define above.

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A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

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Q: For each point a∈R of the ring ≺R,+,⋅≻, the mapping θa:≺R,+,⋅≻→≺R,+,⋅≻ defined by θa(x)=a⋅x,  ∀x∈R…

A: Consider R = Z6 Then, we know obviously that (R, +, . ) is a ring. Consider a = 2.So,  θ2(x)=2x

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A: let alpha and beta are in R

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A: This question is about application number theorem

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Q: Consider the ring C[0, 1] of continuous functions f [0, 1] → R under the usual pointwise operations.…

A: The ring  of continuous functions .