Let R be the ring of all real-valued functions defined on the real line.
(a) Find all the zero-divisors in R.
(b) Find all the units in R.
Can someone please help me out a bit, thanks.
Of course, "units" are precisely those members that have multiplicative inverses and the "zero divisors" those that do not.
Of course, the real valued functions form a ring with two different kinds of "multiplication"- f(x)g(x) and f(g(x)). Which do you mean?
Most probably he means the first one since the second one isn't, in a strict fashion, a ring
since functions composition may not yield a function defined in the WHOLE real numbers.
For example, the functions $\displaystyle f(x):=x^2\,,\,\,g(x)=x $ are defined on the real line, but $\displaystyle g\circ f$ is only defined
in the half line $\displaystyle [0,\infty)$ .
Tonio
Not really, in $\displaystyle \mathbb{Z} $ 2 is not a unit or a zero divisor, which is why I gave the fact that in this case it is true as a hint.
Isn't one of them the identity? I don't see how the composition is only defined there. Besides it's easier to see that composition fails to distribute over the sum when the functions involved are non-linear.For example, the functions are defined on the real line, but is only defined
in the half line .
for any $\displaystyle f \in R$ let $\displaystyle Z(f)=\{a \in \mathbb{R}: \ f(a)=0 \}.$ then $\displaystyle \{ f \in R: \ Z(f) \neq \emptyset \}$ is the set of zero divisors of $\displaystyle R$ and $\displaystyle \{f \in R: \ Z(f) = \emptyset \}$ is the set of units of $\displaystyle R$.