Finite cring is cohomologous to direct product of crings of prime power order

Statement
Suppose $$C$$ is a finite cring. Then, there is a finite cring $$C_1$$ that is cohomologous to $$C$$ and that is expressible as a direct product of crings of prime power order corresponding to the maximal prime powers dividing the order of $$C$$.

Related facts

 * Equivalence of definitions of finite nilpotent group
 * Equivalence of definitions of finite nilpotent Moufang loop
 * Finite ring is internal direct product of its Sylow subrings