Originally Posted by

**matt.qmar** I think I made some progress? not sure if it is valid...

Since $\displaystyle <a>\ = R$, then we can write each element $\displaystyle r \in R$ as $\displaystyle r = ac$ for some $\displaystyle c \in R$.

But since $\displaystyle a$ itself is in $\displaystyle R$, then $\displaystyle ac = a$ for some $\displaystyle c \in r$.

Since $\displaystyle R$ is commutative, $\displaystyle ac = ca = a$

Does this mean that we have a unity? How do we prove that $\displaystyle c$ works for every element of $\displaystyle R$?

Once we can show we have a unity, I think the rest is done, because:

if $\displaystyle r \in R$, then $\displaystyle <r> = R$. So each element of $\displaystyle R$ can be written as $\displaystyle rd$ for some $\displaystyle d \in R$. Since $\displaystyle 1_R \in R$, $\displaystyle 1_R = rd$ for some $\displaystyle d \in R$. Since $\displaystyle R$ is commutative, $\displaystyle 1_R = rd = dr$, so by the definition of unique inverses, each element has an inverse, so each element is a unit, so $\displaystyle R$ is a field.

Can anyone look over this argument and verify it/spot a problem? I'd be very grateful, thanks!