Let W be the set of all continuous fuctions 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)=f(x)g(x). Verify that W is a commutative ring. Does W have unity? Why or why not?
To verify that W is a commutative ring, we need to check that it satisfies several properties:
1. Addition is closed: For any two functions f and g in W, the sum (f+g)(x) = f(x) + g(x) is also a real-valued function.
2. Addition is commutative: For any two functions f and g in W, (f+g)(x) = f(x) + g(x) = g(x) + f(x) = (g+f)(x).
3. Addition is associative: For any three functions f, g, and h in W, ((f+g)+h)(x) = (f+g)(x) + h(x) = (f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x)) = (f+(g+h))(x).
4. Commutative multiplication: For any two functions f and g in W, (f*g)(x) = f(x) * g(x) = g(x) * f(x) = (g*f)(x).
5. Multiplication is closed: For any two functions f and g in W, the product (f*g)(x) = f(x) * g(x) is also a real-valued function.
6. Multiplication is associative: For any three functions f, g, and h in W, ((f*g)*h)(x) = (f*g)(x) * h(x) = (f(x) * g(x)) * h(x) = f(x) * (g(x) * h(x)) = (f*(g*h))(x).
7. Distributive property: For any three functions f, g, and h in W, (f*(g+h))(x) = f(x) * (g(x) + h(x)) = (f(x) * g(x)) + (f(x) * h(x)) = (f*g)(x) + (f*h)(x).
Hence, we have verified that W is a commutative ring.
However, W does not have a unity/identity element. In a ring, the unity element, denoted as 1, is an element such that for any element a in the ring, a*1 = 1*a = a. The unity element is unique in a ring.
In the given set W of continuous functions from R to R, there is no constant function that can serve as the unity element. By definition, a unity element should be a constant function such that for any function f in W, f*1 = 1*f = f. However, for any non-constant function f in W, f(x) ≠ c for some constant c, which means there is no constant function that can satisfy the requirements of a unity element. Therefore, W does not have a unity element.