zero divisor

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English

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Alternative forms

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Noun

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zero divisor (plural zero divisors)

  1. (algebra, ring theory) An element a of a ring R for which there exists some nonzero element xR such that either ax = 0 or xa = 0.
    An idempotent element of a ring is always a (two-sided) zero divisor, since .
    • 1984, J. B. Srivastava, “23: Projective Modules, Zero Divisors, and Noetherian Group Algebras”, in Dinesh N. Manocha, editor, Algebra and its Applications, CRC Press, page 170:
      Linnell [25, 1977] proved that if G is a torsion-free abelian by locally finite by super-solvable group and K is any field, then K[G] has no nontrivial zero divisors.
    • 1989, K. D. Joshi, Foundations of Discrete Mathematics, New Age International, page 390:
      In the ring of integers, there are no zero divisors except 0. In a ring obtained from a Boolean algebra, on the other hand, every element except the identity is a zero-divisor.
      The concept of a zero-divisor is intimately related to cancellation law as we see n the following proposition.
      1.7 Proposition: Let be a ring and . Then for all , either of the equations or implies if and only if is not a zero divisor. In other words, cancellation by an element is possible iff it is not a zero-divisor.
    • 2010, Mitsuo Kanemitsu, “The Number of Distinct 4-Cycles and 2-Matchings of Some Zero Divisor Graphs”, in Masami Ito, Yuji Kobayashi, Kunitaka Shoji, editors, Automata, Formal Languages and Algebraic Systems: Proceedings of AFLAS 2008, World Scientific, page 63:
      In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors.
  2. (algebra, ring theory) A nonzero element a of a ring R for which there exists some nonzero element xR such that either ax = 0 or xa = 0.
    • 2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234:
      If is an integral domain, that is, has no zero divisors, then also has no zero divisors.
    • 2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi:
      An element of a ring is called a zero-divisor if and or for some ; if is neither 0 nor a zero-divisor, it is said to be regular (see Section 7.1). A non-trivial ring without zero-divisors is called an integral domain; this term is not taken to imply commutativity.
    • 2009, Victor Shoup, A Computational Introduction to Number Theory and Algebra, 2nd edition, Cambridge University Press, page 171:
      If and are non-zero elements of such that , then a and are both called zero divisors. If is non-trivial and has no zero divisors, then it is called an integral domain. Note that if is a unit in , it cannot be a zero divisor (if , then multiplying both sides of this equation by yields .

Usage notes

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  • The two definitions differ according to whether or not 0 is considered a zero divisor.
  • Related terminology:
    • An element (resp. nonzero element) such that is called a left zero divisor.
    • An element (resp. nonzero element) such that is called a right zero divisor.
    • An element that is both a left zero divisor and a right zero divisor is called a two-sided zero divisor.
  • 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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  • (antonym(s) of any element whose product with some nonzero element is zero): regular element

Hyponyms

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Derived terms

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Translations

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See also

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Further reading

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