Question

Let W be the set of all continuous functions from the set of real numbers R to R. For f,g in W define f+g by (f+g)(x)=f(x)+g(x) and (f*g)(x)= f(x)g(x). Verify that W is a commutative ring. Does W have unity? Why or why not?

Answer 1

A ring has to be equipped with two operations, generally referred to as addition and multiplication.
Addition has been defined as:
f+g : (f+g)(x)=f(x)+g(x)
f*g : (f*g)(x)= f(x)g(x)

In order that W is a ring, it has to satisfy the following axioms:
Addition:
1. closure
2. associativity
3. existence of zero element
4. inverse of addition
5. commutativity
Multiplication:
1. closure
2. associativity
Distributivity
1. distributivity (left), i.e.
∀ a,b,c∈W, a(b+c)=ab+ac
2. distributivity (right),
∀ a,b,c,isin;W, (a+b)c=ac+bc

If you can demonstrate that the given operations satisfy all of the above, the W is a ring.

Unity(identity) is designated as "1", and has the property of
1a=a1=a ∀a∈W.
Note that the elements of W are all functions, so unity should also be a function, say, u(x) such that
u(x).f(x)=f(x).u(x)=f(x)

Based on this information, you should be able to get started (and finish). Post your answer if you wish for critique.