How can I prove if $R$ is a domain where every submodule is a summand, then it is a field? Suppose $R$ is a domain with the property that, for $R$-modules, every
submodule is a summand. 
I would like to show $R$ is a field.
Stating the definitions I know that for any submodule $A$ there exsists a summand $B$ where $A \oplus B = R$ and $A \cap B=0$.
Also $\forall a,b \in A,B$, $ab \neq0.$
However I am not sure the first step to take to show that every element of $R$ is a unit.
Thanks

Let $A$ be an ideal of $R$ and $B$ such that $A \oplus B = R, A \cap B = 0$. Take $a \in A, b \in B$. Then $ab \in A \cap B$. Now use that $R$ is a domain to conclude that $A, B$ are trivial.
