Associative and Distributive Properties of Multiplication

Let F be the set of all functions f : RR. We know that <F, +> is an abelian group under the usual function addition, (f + g)(x) = f(x) + g(x). We define multiplication on F by (fg)(x) = f(x)g(x). That is, fg is the function whose value at x is f(x)g(x). Show that the multiplication defined on the set F satisfies axioms R2 and R3 for a ring.

R2: Multiplication is associative.
R3: For all a, b, c Є R, the left distributive law, a.(b+c) = (ab) +(ac) and the right distributive law (a+b)c = (ac) + (bc) hold.

*Note that R is the ring.

See attached file for full problem description.

Attached is a problem from an undergraduate course in Abstract Algebra. The book we use is titled "A First Course in Abstract Algebra" by John B. Fraleigh. We have just started Ring and Field Theory. If you are able to solve the following problem, please detail any theorems or definitions in your answers.

Associative and distributive properties of multiplication for a ring are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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