See zero divisor on Wiktionary
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"sense": "any element whose product with some nonzero element is zero",
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"text": "An idempotent element e#x5C;ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)#x3D;0#x3D;(1-e)e.",
"type": "example"
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{
"ref": "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:",
"text": "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.",
"type": "quote"
},
{
"ref": "1989, K. D. Joshi, Foundations of Discrete Mathematics, New Age International, page 390:",
"text": "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.\nThe concept of a zero-divisor is intimately related to cancellation law as we see n the following proposition.\n1.7 Proposition: Let R be a ring and x#x5C;inR. Then for all y,x#x5C;inR, either of the equations xy#x3D;xz or yx#x3D;zx implies y#x3D;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.",
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"ref": "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:",
"text": "In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors.",
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"An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0."
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"text": "If R is an integral domain, that is, has no zero divisors, then R#x5B;x#x5D; also has no zero divisors.",
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"ref": "2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi:",
"text": "An element a of a ring is called a zero-divisor if a#x5C;ne 0 and ab#x3D;0 or ba#x3D;0 for some b#x5C;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.",
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"ref": "2009, Victor Shoup, A Computational Introduction to Number Theory and Algebra, 2nd edition, Cambridge University Press, page 171:",
"text": "If a and b are non-zero elements of R such that ab#x3D;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#x3D;0, then multiplying both sides of this equation by a#x7B;-1#x7D; yields b#x3D;0.",
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"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."
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"ref": "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:",
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