zero divisor

plural zero divisors · alternative zero-divisor

1. ring theoryalgebraAn 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#92;ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)#61;0#61;(1-e)e.

   3 more examples

   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.

   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

   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#92;inR. Then for all y,x#92;inR, either of the equations xy#61;xz or yx#61;zx implies y#61;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.

   1989, K. D. Joshi, Foundations of Discrete Mathematics, New Age International, page 390

   In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors.

   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

2. ring theoryalgebraA nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

   If R is an integral domain, that is, has no zero divisors, then R#91;x#93; also has no zero divisors.

   2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234

   2 more examples

   An element a of a ring is called a zero-divisor if a#92;ne 0 and ab#61;0 or ba#61;0 for some b#92;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.

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

   If a and b are non-zero elements of R such that ab#61;0, then aandb 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#61;0, then multiplying both sides of this equation by a#123;-1#125; yields b#61;0.

   2009, Victor Shoup, A Computational Introduction to Number Theory and Algebra, 2nd edition, Cambridge University Press, page 171