Zero divisor

English

Alternative forms

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Noun

1. Template:Lb An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

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2. Template:Lb A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.
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Usage notes

• The two definitions differ according to whether or not 0 is considered a zero divisor.
  • The definition that includes 0 is the one preferred by Template:W. (See reference cited in Template:Pedia)
  • Additionally, 0 may be called the Template:M.
• Related terminology:
  • An element (resp. nonzero element) $a$ such that $\mathrm{\exists }x\in R:x\ne 0\wedge ax=0$ is called a Template:M.
  • An element (resp. nonzero element) $a$ such that $\mathrm{\exists }x\in R:x\ne 0\wedge xa=0$ is called a Template:M.
  • An element $a$ that is both a left zero divisor and a right zero divisor is called a Template:M.
• Thus, a zero divisor can be (and often is) defined as any element that is either a left zero divisor or a right zero divisor.
• The term zero divisor is most relevant in the context of commutative rings (where the left–right distinction is not made).

Antonyms

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Hyponyms

• Template:Sense Template:M
• Template:Sense Template:L, Template:L, Template:L, Template:L

Derived terms

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Translations

• Finnish: Template:T
• French: Template:T+

See also

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Further reading

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• Zero divisor on Template:W
• Zero Divisor on Template:W