zero divisor

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zero divisor

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Alternative forms

• zero-divisor

Noun

zero divisor (plural zero divisors)

1. (algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

   An idempotent element ${\displaystyle e\neq 1}$ of a ring is always a (two-sided) zero divisor, since ${\displaystyle e(1-e)=0=(1-e)e}$.

   • 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 ${\displaystyle R}$ be a ring and ${\displaystyle x\in R}$. Then for all ${\displaystyle y,x\in R}$, either of the equations ${\displaystyle xy=xz}$ or ${\displaystyle yx=zx}$ implies ${\displaystyle y=z}$ if and only if ${\displaystyle x}$ 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 x ∈ R such that either ax = 0 or xa = 0.
   • 2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234:

     If ${\displaystyle R}$ is an integral domain, that is, has no zero divisors, then ${\displaystyle R}$ also has no zero divisors.

   • 2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi:

     An element ${\displaystyle a}$ of a ring is called a zero-divisor if ${\displaystyle a\neq 0}$ and ${\displaystyle ab=0}$ or ${\displaystyle ba=0}$ for some ${\displaystyle b\neq 0}$; if ${\displaystyle a}$ 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 ${\displaystyle a}$ and ${\displaystyle b}$ are non-zero elements of ${\displaystyle R}$ such that ${\displaystyle ab=0}$, then ${\displaystyle }$a and ${\displaystyle b}$ are both called zero divisors. If ${\displaystyle R}$ is non-trivial and has no zero divisors, then it is called an integral domain. Note that if ${\displaystyle a}$ is a unit in ${\displaystyle R}$, it cannot be a zero divisor (if ${\displaystyle ab=0}$, then multiplying both sides of this equation by ${\displaystyle a^{-1}}$ yields ${\displaystyle b=0}$.

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 Bourbaki. (See reference cited in zero divisor on Wikipedia.Wikipedia )
  • Additionally, 0 may be called the trivial zero divisor.
• Related terminology:
  • An element (resp. nonzero element) ${\displaystyle a}$ such that ${\displaystyle \exists x\in R:x\neq 0\land ax=0}$ is called a left zero divisor.
  • An element (resp. nonzero element) ${\displaystyle a}$ such that ${\displaystyle \exists x\in R:x\neq 0\land xa=0}$ is called a right zero divisor.
  • An element ${\displaystyle a}$ 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

• (antonym(s) of "any element whose product with some nonzero element is zero"): regular element

Hyponyms

• (any element whose product with some nonzero element is zero): trivial zero divisor
• (both senses): exact zero divisor, left zero divisor, right zero divisor, two-sided zero divisor

Derived terms

• zero divisor graph

Translations

See also

• annihilator
• integral domain
• nilpotent

Further reading

• Annihilator (ring theory) on Wikipedia.Wikipedia
• Topological divisor of zero on Wikipedia.Wikipedia
• Zero-divisor graph on Wikipedia.Wikipedia
• Zero-product property on Wikipedia.Wikipedia
• Zero divisor on Encyclopedia of Mathematics
• Zero Divisor on Wolfram MathWorld