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| Title of object: |
zero divisor |
| Canonical Name: |
ZeroDivisor |
| Type: |
Definition |
| Created on: |
2002-07-06 14:23:00 |
| Modified on: |
2005-04-20 19:01:41 |
| Classification: |
msc:13G05 |
| Defines: |
left zero divisor, right zero divisor, regular element |
Preamble:
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Content:
Let $a$ be a nonzero element of a ring $R$.
The element $a$ is a {\em left zero divisor} if there exists a nonzero element $b \in R$ such that $a \cdot b = 0$. Similarly, $a$ is a {\em right zero divisor} if there exists $c \in R$ such that $c \cdot a = 0$.
The element $a$ is said to be a {\em zero divisor} if it is both a left and right zero divisor. A nonzero element $a \in R$ is said to be a {\em regular element} if it is neither a left nor a right zero divisor.
{\bf 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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