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Does the ring of continuous functions over the real numbers have no zero divisors? If no 0 divisor, how can I prove it? Else, what is a counter example?
A zero divisor in this context is a non-zero function f(x) such that there exists a non-zero function g(x) where the product of f(x) and g(x) equals zero for all values of x in the domain.
To identify if a function is a zero divisor, you can check if there exists a non-zero function g(x) such that the product of f(x) and g(x) is equal to zero for all values of x in the domain. If such a function exists, then f(x) is a zero divisor.
No, not all continuous functions over R are zero divisors. For example, the function f(x) = 1 is not a zero divisor because there does not exist a non-zero function g(x) where the product of f(x) and g(x) is equal to zero for all values of x in the domain.
Zero divisors have significance in abstract algebra and ring theory. They play a crucial role in understanding the properties and structure of rings, and can also be used to prove theorems and solve problems related to rings and fields.
Yes, a continuous function over R can have more than one zero divisor. For example, the function f(x) = x^2 has two zero divisors: g1(x) = x and g2(x) = -x. Both g1(x) and g2(x) are non-zero functions, and their product with f(x) equals zero for all values of x in the domain.