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?
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.
To verify that W is a commutative ring, we need to show that the set W satisfies the properties of a commutative ring: addition, subtraction, multiplication, commutativity, associativity, distributivity, and the existence of additive and multiplicative identities.
1. Closure under Addition:
For any two functions f, g in W, (f+g)(x) = f(x) + g(x) is a continuous function since the sum of continuous functions is continuous. Thus, W is closed under addition.
2. Closure under Subtraction:
For any two functions f, g in W, (f-g)(x) = f(x) - g(x) is a continuous function since the difference of continuous functions is continuous. Thus, W is closed under subtraction.
3. Closure under Multiplication:
For any two functions f, g in W, (f*g)(x) = f(x) * g(x) is a continuous function since the product of continuous functions is continuous. Thus, W is closed under multiplication.
4. Commutativity of Addition:
For any two functions f, g in W, (f+g)(x) = f(x) + g(x) = g(x) + f(x) = (g+f)(x). Hence, addition is commutative in W.
5. Associativity of Addition:
For any three functions f, g, 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). Hence, addition is associative in W.
6. Distributivity:
For any three functions f, g, h in W, ((f*g)+h)(x) = (f*g)(x) + h(x) = f(x)*g(x) + h(x), and (f*(g+h))(x) = f(x) * (g(x) + h(x)). By the distributive property of real numbers, we know that f(x)*g(x) + f(x)*h(x) = f(x) * (g(x) + h(x)). Therefore, ((f*g)+h)(x) = (f*(g+h))(x), and distributivity holds in W.
7. Existence of Additive and Multiplicative Identities:
The additive identity in W is the function 0(x) = 0 for all x, as (f+0)(x) = f(x) + 0(x) = f(x) for any f in W.
However, W does not have a multiplicative identity. To have a multiplicative identity, we would need a function e(x) such that (f*e)(x) = f(x) for all f in W. However, there is no continuous function that satisfies this condition for all functions in W.
Therefore, W is a commutative ring but does not have a multiplicative identity.
To verify that W is a commutative ring, we need to check that it satisfies certain properties.
1. Closure under addition: Let f, g be continuous functions in W. Then (f + g)(x) = f(x) + g(x), which is also a continuous function since the sum of continuous functions is continuous. Therefore, W is closed under addition.
2. Closure under multiplication: Let f, g be continuous functions in W. Then (f * g)(x) = f(x) * g(x), which is also a continuous function since the product of continuous functions is continuous. Therefore, W is closed under multiplication.
3. Commutativity of addition: Let f, g be continuous functions in W. Then (f + g)(x) = f(x) + g(x) = g(x) + f(x) = (g + f)(x). Therefore, addition is commutative in W.
4. Associativity of addition: Let f, g, h be continuous functions in W. Then ((f + g) + h)(x) = (f + g)(x) + h(x) = (f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x)) = f(x) + (g + h)(x) = (f + (g + h))(x). Therefore, addition is associative in W.
5. Additive identity: There exists a function 0 in W such that for any f in W, (f + 0)(x) = f(x) + 0(x) = f(x) for all x in R. This identity function is simply the constant function 0, where 0(x) = 0 for all x in R.
6. Additive inverse: For any f in W, there exists a function -f in W such that (f + (-f))(x) = f(x) + (-f)(x) = 0 for all x in R. The function -f(x) = -f(x) for all x in R serves as the additive inverse of f.
Since W satisfies all the properties above, it is a commutative ring.
Regarding the existence of unity, a ring has a unity if there exists a multiplicative identity element. In this case, a function 1 in W would need to satisfy (f * 1)(x) = f(x) for all f in W and all x in R. However, there is no function that can serve as the multiplicative identity in W since for any non-constant function f, (f * 1)(x) = f(x) * 1(x) = f(x) * 1 = f(x) ≠ f(x) for all x in R. Therefore, W does not have a unity.