A finite ring without zero divisors has a neutral element for the multiplication.

Question

$(R,+,\cdot)$ is a finite ring without zero divisors. Prove that $R$ has a neutral element for the multiplication.

Can someone give a hint or something? Thanks.

Answer 1

HINT:

Fix $r\in R$, $r\neq 0$, and define $f:R\to R$ as $f(x)=rx$. Use the finite property and the fact that $R$ has no zero divisors to find an element $e\in R$ such that $r\cdot e=r$ (remember that every function $g\in A^A$ is surjective iff it's injective, when $A$ is finite).

Then try to prove that $a$ is actually your identity.