Finite ring is internal direct product of its Sylow subrings

Statement
Suppose $$R$$ is a finite ring, i.e., a non-associative ring (here non-associative means not necessarily associative) whose underlying set is finite, of order $$n$$ with prime factorization:

$$\! n = p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}$$

where the $$p_i$$s are distinct primes. Then, we can write $$R$$ as an internal direct product:

$$R = R_1 \times R_2 \times \dots \times R_r$$

where each $$R_i$$ is a subring of $$R$$ of order $$p_i^{k_i}$$ (and is in fact also an ideal of $$R$$). In fact, each $$R_i$$ is uniquely determined by $$R$$ and can be defined as the set $$\{ (n/p_i^{k_i})x \mid x \in R \}$$.

Related facts

 * Equivalence of definitions of finite nilpotent group
 * Equivalence of definitions of finite nilpotent Moufang loop