Exercise 6, § 18

Let A be a ring such that for every $a\in A$, there exists an integer $n(a)>1$ such that $a^{n(a)}=a$. Prove that A is isomorphic to a product of finitely many reduced commutative algebraic algebras over finite fields. (Observe that A is annihilated by an integer $n$, and then that for every prime number $p$, the $p$-component $A_p$ of the additive group A is a two-sided ideal annihilated by $p$. Apply Exercise 4, b).) Also prove the converse.