Existence of an identity element in a finite ring

#ring_theory

Suppose $R$ is a finite ring with no nontrivial zero-divisors. Prove that $R$ contains an element $1$ satisfying $1 \cdot a = a \cdot 1 = a$ for all $a \in R$ .

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Suppose $R$ is a finite ring with no nontrivial zero-divisors. Prove that $R$ contains an element $1$ satisfying $1\cdot a=a\cdot 1=a$ for all $a\in R$.