Zero-divisor

Let $𝑅$ be a ring. A left (right) zero-divisor is an element $𝑧 \in 𝑅$ which sends some nonzero element to zero when multiplying on the left (right), #m/def/ring i.e. $𝑧 𝑎 = 0$ ( $𝑎 𝑧 = 0$ ) for some $𝑎 \in 𝑅$ with $𝑎 \neq 0$ .

As morphisms

Let $𝑅 ――$ denote the multiplicative monoid of a ring $𝑅$ viewed as a category. Then $𝑎 \in 𝑅$ is

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

If we view $Λ (𝑎)$ and $P (𝑎)$ as functions on $𝑅$ , then $𝑎 \in 𝑅$ is¹

a left zero-divisor iff $Λ (𝑎)$ is not injective;
a right zero-divisor iff $P (𝑎)$ is not injective.

Zero-divisor

As morphisms

See also

Footnotes