I have the following here.
For each of the following rings $R$, decide whether it is an integral domain. If
it is, prove so. Otherwise, explain why not with the aid of a counterexample.
a) $R=\text{Map}(\mathbb{R},\mathbb{R})$.
b) $S=\text{Map}(\mathbb{R},\mathbb{R})/I$, $I=\{f\in\text{Map}(\mathbb{R},\mathbb{R})|f(1)=0\}$.
I have the following definition of an integral domain:
"An integral domain is a commutative ring $R$ such that:
i) $R$ has an identity $\mathbf{1}\neq 0$
ii) For all $a,b\in R$, if $ab=0$, then either $a=0$ or $b=0$.
Additionally, an integral domain has no zero divisors."
I think a) is an integral domain. You can't multiply two non zero elements to get $0$. I am not sure how to rigorously prove this though. It just seems so obvious… For non zero elements $a$ and $b$, if I multiply $a$ and $b$, we get $ab=0$ but you can never get 0 for this ring for any two non zero elements. How do I show this rigorously though? It seems trivial.
b) Should not be an integral domain. $f(1)=0$ means that regardless of what numbers you multiply, there will always be some kind of $0$ divisor.
I think I have the jist of this but I am not sure how to word this properly. Can someone help out here?
Best Answer
Hints.
a) $R$ is not an integral domain. (One can easily find two non-zero functions whose product is zero everywhere.)
b) $R/I$ is isomorphic to $\mathbb R$. In order to show this associate to every function $f\in R$ its value at $1$, that is, $f\mapsto f(1)$ and then find the kernel and image of this map.