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zero divisor

Zero divisor

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zero divisor

名詞

zero divisor (複数形 zero divisors)

(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 e ≠ 1 {\displaystyle e\neq 1} $e\neq 1$ of a ring is always a (two-sided) zero divisor, since e ( 1 − e ) = 0 = ( 1 − e ) e {\displaystyle e(1-e)=0=(1-e)e} $e(1-e)=0=(1-e)e$ .
- 1984, J. B. Srivastava, 23: Projective Modules, Zero Divisors, and Noetherian Group Algebras, 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 R {\displaystyle R} $R$ be a ring and x ∈ R {\displaystyle x\in R} $x\in R$ . Then for all y , x ∈ R {\displaystyle y,x\in R} $y,x\in R$ , either of the equations x y = x z {\displaystyle xy=xz} $xy=xz$ or y x = z x {\displaystyle yx=zx} $yx=zx$ implies y = z {\displaystyle y=z} $y=z$ if and only if x {\displaystyle x} $x$ 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, 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.
(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, Springer, 2nd Edition, page 234,
  If R {\displaystyle R} $R$ is an integral domain, that is, has no zero divisors, then R [ x ] {\displaystyle R[x]} $R[x]$ also has no zero divisors.
- 2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi,
  An element a {\displaystyle a} $a$ of a ring is called a zero-divisor if a ≠ 0 {\displaystyle a\neq 0} $a\neq 0$ and a b = 0 {\displaystyle ab=0} $ab=0$ or b a = 0 {\displaystyle ba=0} $ba=0$ for some b ≠ 0 {\displaystyle b\neq 0} $b\neq 0$ ; if a {\displaystyle a} $a$ 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, Cambridge University Press, 2nd Edition, page 171,
  If a {\displaystyle a} $a$ and b {\displaystyle b} $b$ are non-zero elements of R {\displaystyle R} $R$ such that a b = 0 {\displaystyle ab=0} $ab=0$ , then a and b {\displaystyle b} $b$ are both called zero divisors. If R {\displaystyle R} $R$ is non-trivial and has no zero divisors, then it is called an integral domain. Note that if a {\displaystyle a} $a$ is a unit in R {\displaystyle R} $R$ , it cannot be a zero divisor (if a b = 0 {\displaystyle ab=0} $ab=0$ , then multiplying both sides of this equation by a − 1 {\displaystyle a^{-1}} $a^{-1}$ yields b = 0 {\displaystyle b=0} $b=0$ .

使用する際の注意点

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.)
- Additionally, 0 may be called the trivial zero divisor.
Related terminology:
Thus, a zero divisor can be (かつ 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).

下位語

(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

参考

Zero divisor

出典:『Wikipedia』 (2011/05/06 04:04 UTC 版)

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zero-divisor

「zero divisor」の部分一致の例文検索結果

zero-divisor

zero divisor

zero divisor

Zero divisor

zero divisor

名詞

使用する際の注意点

下位語

参考

Further reading

Zero divisor