Exercise in abstract algebra Assume that $M$ is a set, and that $R$ is a ring. Let $\cal{F}$ be the collection all functions $M\rightarrow{R}$. Prove that the following two statements are equivalent.


*

*$\cal{F}$ is a field.

*$M$ is a singleton (i.e. consisting one element), and $R$ is a field.
The implication 2. $\implies$ 1. is straightforward. If $M=\{m\}$, one identifies any particular element $r\in{R}$ with the function $f(m) = r$, and checks that $\cal{F}$ satisfies the field axioms. But I am stuck on the implication 1. $\implies$ 2.

Suppose M has at least two distinct elements : $m_1$ and $m_2$.
Then, consider $f :M\to R$ such that $f(m_1)=1$ and $f(m)= 0$ otherwise; and $g:M\to R$ such that $g(m_2)=1$ and $g(m) =0$ otherwise.
Then, as functions (that is, as elements of $F$), $f$ and $g$ are non zero but $fg=0$.
So $F$ can't be a field.
