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Let $a$ be a nonzero element of a ring $R$ .
The element $a$ is a left zero divisor if there exists a nonzero element $b \in R$ such that $a \cdot b = 0$ . Similarly, $a$ is a right zero divisor if there exists a nonzero element $c \in R$ such that $c \cdot a = 0$ .
The element $a$ is said to be a zero divisor if it is both a left and right zero divisor. A nonzero element $a \in R$ is said to be a regular element if it is neither a left nor a right zero divisor.
Example: Let $R = \mathbb{Z}_6$ . Then the elements $2$ and $3$ are zero divisors, since $2 \cdot 3 \equiv 6 \equiv 0 \pmod 6$ .
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