Ring theory MOC

Zero-divisor

Let 𝑅 be a ring. A left (right) zero-divisor is an element 𝑧 ∈𝑅 which sends some nonzero element to zero when multiplying on the left (right), ring i.e. 𝑧𝑎 =0 (𝑎𝑧 =0) for some 𝑎 ∈𝑅 with 𝑎 ≠0.

As morphisms

Let 𝑅―― denote the multiplicative monoid of a ring 𝑅 viewed as a category. Then 𝑎 ∈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 ∈𝑅 is¹

• a left zero-divisor iff Λ(𝑎) is not injective;

• a right zero-divisor iff P(𝑎) is not injective.

See also

• Unit

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Footnotes

1. 2009. Algebra: Chapter 0,§III.1.2, ¶1.9, p. 122 ↩