Right Zero Divisor of Commutative Ring is Left Zero Divisor

Theorem

Let $\struct {R, +, \circ}$ be a commutative ring.

Let $x \in R$ be a right zero divisor of $R$.

Then $x$ is also a left zero divisor of $R$.

Proof

Let $x \in R$ be a right zero divisor of $R$.

Let $0_R$ denote the zero of $R$.

Then:

\(\ds \exists y \in R^*: \, \)

\(\ds x \circ y\)

\(=\)

\(\ds 0_R\)

Definition of Left Zero Divisor

\(\ds \leadsto \ \ \)

\(\ds y \circ x\)

\(=\)

\(\ds 0_R\)

Definition of Commutative Ring

Hence $x$ is a left zero divisor of $R$ by definition.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisor of zero
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisor of zero