zero divisor
Noun
This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.002
Noun
zero divisor (plural 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\ne 1 of a ring is always a (two-sided) zero divisor, since 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 be a ring and x\in R. Then for all y, x\in R, either of the equations xy=xz or yx=zx implies y=z if and only if 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 is an integral domain, that is, has no zero divisors, then R[x] also has no zero divisors.
- 2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi ↗,
- An element a of a ring is called a zero-divisor if a\ne 0 and ab = 0 or ba = 0 for some b\ne 0; if 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 and b are non-zero elements of R such that ab = 0, then a and b are both called zero divisors. If R is non-trivial and has no zero divisors, then it is called an integral domain. Note that if a is a unit in R, it cannot be a zero divisor (if ab = 0, then multiplying both sides of this equation by a^{-1} yields b=0.
- 2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, Springer, 2nd Edition, page 234 ↗,
- (any element whose product with some nonzero element is zero) regular element
- French: diviseur de zéro
This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.002