Let $R$ be a ring such that $1\neq 0$ and $1+1=0$. Suppose that $R$ contains a subring isomorphic to $\mathbb{F}_2^n.$ Then, there are rings $R_1,\ldots, R_n$ for which $R\cong R_1\times R_2\times\cdots R_n$.
I hope someone give me a help with the solution, please.
The condition $1+1=0$ implies that $x+x=0$ for every $x$, and this in turn can be used to show that $R$ is actually a vector space over $\Bbb F_2$ (if you define scalar multiplication in the only possible way, namely $0\cdot x =0$ and $1\cdot x=x$).
The other hypothesis implies that the dimension of this vector space is at least $n$, and the result follows (partition a basis into $n$ disjoint nonempty subsets...)