Existence of submodules

Question

Show that if $R$ has an identity and $A$ is a $R$-module, then there are submodules $B$ and $C$ of $A$ such that $B$ is unitary, $RC = 0$ and $A = B ⊕ C.$

Give my hint, I don't know how to start.

Answer 1

I'm assuming that a unitary $R$-module $A$ is precisely one where $1a=a$ for all $a\in A$?

If $A$ is not unitary, try $B:=\{b\in A : 1b=b\}$ and $C:=\{c\in A : 1c=0\}$. I'll leave it to you to check that $B,C$ are submodules, $RC=0$, $B$ is unitary, and $B\cap C=0$.

To see that $A=B+C$, I'll give you a hint: $a=1a+(a-1a)$.