This question wants me to use a previous result, that if $e$ is a central idempotent in $R$ then $R \cong Re \oplus R(1-e)$. I was thinking about maybe some sort of induction would help here but I can't see exactly how I am going to get to the result I want.
I would like to know how to do the problem this way, because this is the way the book recommends. However, I also have heard that this problem can be done another way, by realizing that $R$ is a vector space over $Z_2$ and there has a finite basis, and i am under the impression that the result is not far away knowing this, but I can't quite connect the dots.
If somebody could guide me along these two different routes of solving this problem I would really appreciate it! Thanks!
A boolean ring is always commutative: $$a+b=(a+b)^2 =a^2 +b^2 + ab + ba = a+b+ab+ba$$ Hence $ab+ba=0$. Applying the same with $a=b=x$ yields $x+x=0$, so $x=-x$ for all $x$, and we have $ba=-ab=ab$.
And, by definition, also every element is idempotent, so we can arbitrarily pick a nontrivial element $e$, until there is such, and use induction and the previously proved claim.
By the above result ($x+x=0$), we indeed get a $\Bbb Z_2$ vector space structure in $R$, and since it's finite, it has a finite basis. All you have to verify is that multiplication of $R$ coincides with the coordinatewise multiplication once a basis is fixed.