Here is the problem: Let $R$ be a ring of real valued functions over $[0,1]$. Let $S$ be the set of functions all with an infinite number of zeroes. Determine whether or not $S$ is a subring of $R$.
I am pretty sure that $S$ is not a subring of $R$. So I'm trying to find a counterexample. For some reason, I am having trouble finding two functions that both have an infinite number of zeroes and whose sum/difference has a finite number of zeroes. Any help?
Let $f$ be zero on rational numbers and $1$ on irrational numbers; let $g$ be zero on irrational numbers and $1$ on rational numbers. Then consider $f+g$. Clearly $f, g$ are in $S$ but $f+g = 1$. (Note: $f$ and $g$ are not continuous, so aren't in $R$ to begin with - I think you might want to say $S$ is set of continuous functions with infinitely many zeroes)