Ring theory MOC

Zero-divisor

Let be a ring. A left (right) zero-divisor is an element which sends some nonzero element to zero when multiplying on the left (right), #m/def/ring i.e. ( ) for some with .

As morphisms

Let denote the multiplicative monoid of a ring viewed as a category. Then is

• a left zero-divisor iff it is not monic;
• a right zero-divisor iff it is not epic.

If we view and as functions on , then is¹

• a left zero-divisor iff is not injective;

• a right zero-divisor iff is not injective.

See also

• Unit

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