# Commutative rings and with zero divisors

$F(R)= {(f: R \rightarrow R })$ is a commutative ring with point wise addition and multiplication.

Find all units and all zero divisors of $F(R)$.

I claimed that $f$ is a unit iff $f(r)\neq 0 ,\forall r\in R$

Then if $f$ is a unit $\exists g \in F(R)$ with $f*g=1$

So $f(r)g(r)=1,\forall r\in R$..

I'm not sure of the zero divisors part does this mean none exist? (Is the proof legit?)

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Consider the functions $f_r$ defined by $f_r(r) = r$ and $f_r(s) = 0$ if $s\neq r.$ What can you conclude about zero divisors –  mike vaiana Oct 25 '13 at 0:53