Left Zero Divisor of Commutative Ring is Right Zero Divisor
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Theorem
Let $\struct {R, +, \circ}$ be a commutative ring.
Let $x \in R$ be a left zero divisor of $R$.
Then $x$ is also a right zero divisor of $R$.
Proof
Let $x \in R$ be a left 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 right 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