Finite Ring with No Proper Zero Divisors is Field

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Theorem

Let $\left({R, +, \circ}\right)$ be a finite non-null ring with no proper zero divisors.


Then $R$ is a field.


Proof

As $R$ is non-null, there is at least one nonzero element in $R$.

Consider the two maps from $R$ to itself, for each nonzero $a \in R$:

$\varphi_R: x \mapsto a \circ x$
$\varphi_L: x \mapsto x \circ a$

By Ring Element is Zero Divisor iff not Cancellable, all nonzero elements in $R$ are cancellable. Thus:

$a \circ x = a \circ y \implies x=y$
$x \circ a = y \circ a \implies x=y$

Therefore, both maps are by definition injective.

By Equivalence of Mappings between Sets of Same Cardinality, the maps are then also surjective.


First, we show that $R$ has a unity.

Since $\varphi_R$ is surjective, any element $b \in R$ can be written in the form:

$b = a \circ x_b$

for some $x_b \in R$.

As $\varphi_L$ is surjective, there is some $x_L \in R$ such that:

$a = x_L \circ a$


We now have:

\(\displaystyle b\) \(=\) \(\displaystyle a \circ x_b\)
\(\displaystyle \) \(=\) \(\displaystyle x_L \circ a \circ x_b\) as $a = x_L \circ a$
\(\displaystyle \) \(=\) \(\displaystyle x_L \circ b\) as $b = a \circ x_b$

Thus, $x_L$ is a multiplicative left identity.

By similar arguments, $R$ also has a multiplicative right identity, which can be denoted $x_R$.

By Left and Right Identity are the Same, it follows that $x_L = x_R$, so they are the unity of $R$.


Now we show that $R$ is a division ring, i.e. that each nonzero element of $R$ has an inverse.

Let $1_R$ denote the unity of $R$.

Since $\varphi_R$ is surjective, it follows that:

$a \circ y_R = 1_R$

for some $y_R \in R$.

Since $\varphi_L$ is surjective, it follows that:

$y_L \circ a = 1_R$

for some $y_L \in R$.

Recalling that the maps were defined for each nonzero $a \in R$, this means that every nonzero element of $R$ has both a left inverse and a right inverse.

By Left Inverse and Right Inverse is Inverse, each nonzero element of $R$ has an inverse, so $R$ is by definition a division ring.


It now follows by Wedderburn's Theorem that $R$ is a field.

$\blacksquare$