A finite unital ring with no zero divisors is a field

Prove that a finite ring with 1 \neq 0 which has no zero divisors is a field.

[We will assume Wedderburn’s Theorem.]

Let R be a finite ring (not necessarily commutative) with 1 \neq 0 having no zero divisors. Let a \in R be nonzero, and define \varphi_a : R \rightarrow R by \varphi_a(r) = ar. Note that if \varphi_a(r) = \varphi_a(s), we have a(r-s) = 0, and since a \neq 0, r = s. Thus \varphi_a is injective. Since R is finite, \varphi_a is surjective. Thus for some b \in R, we have \varphi_a(b) = ab = 1. Similarly, with \psi_a(r) = ra, there exists c \in R such that ca = 1. By a previous theorem, a is a unit in R. Thus R is a division ring.

By Wedderburn’s Theorem, R is commutative; thus R is a field.

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