# zero divisor

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 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 $2\cdot 3\equiv 6\equiv 0\pmod{6}$.

 Title zero divisor Canonical name ZeroDivisor Date of creation 2013-03-22 12:49:59 Last modified on 2013-03-22 12:49:59 Owner cvalente (11260) Last modified by cvalente (11260) Numerical id 9 Author cvalente (11260) Entry type Definition Classification msc 13G05 Related topic CancellationRing Related topic IntegralDomain Related topic Unity Defines left zero divisor Defines right zero divisor Defines regular element