See zero divisor on Wiktionary
Download JSON data for zero divisor meaning in All languages combined (5.9kB)
{
"antonyms": [
{
"sense": "antonym(s) of \"any element whose product with some nonzero element is zero\"",
"word": "regular element"
}
],
"derived": [
{
"_dis1": "46 54",
"word": "zero divisor graph"
}
],
"forms": [
{
"form": "zero divisors",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "zero divisor (plural zero divisors)",
"name": "en-noun"
}
],
"hyponyms": [
{
"_dis1": "49 51",
"sense": "any element whose product with some nonzero element is zero",
"word": "trivial zero divisor"
},
{
"_dis1": "50 50",
"sense": "both senses",
"word": "exact zero divisor"
},
{
"_dis1": "50 50",
"sense": "both senses",
"word": "left zero divisor"
},
{
"_dis1": "50 50",
"sense": "both senses",
"word": "right zero divisor"
},
{
"_dis1": "50 50",
"sense": "both senses",
"word": "two-sided zero divisor"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"_dis1": "46 54",
"word": "annihilator"
},
{
"_dis1": "46 54",
"word": "integral domain"
},
{
"_dis1": "46 54",
"word": "nilpotent"
}
],
"senses": [
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "50 50",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
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"Entry maintenance"
],
"source": "w+disamb"
}
],
"examples": [
{
"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"
},
{
"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": "quotation"
},
{
"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.",
"type": "quotation"
},
{
"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.",
"type": "quotation"
}
],
"glosses": [
"An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0."
],
"id": "en-zero_divisor-en-noun-NveVeim9",
"links": [
[
"algebra",
"algebra"
],
[
"ring",
"ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(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."
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
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"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
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"source": "w"
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{
"_dis": "50 50",
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"examples": [
{
"ref": "2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234",
"text": "If R is an integral domain, that is, has no zero divisors, then R#x5B;x#x5D; also has no zero divisors.",
"type": "quotation"
},
{
"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.",
"type": "quotation"
},
{
"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 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#x3D;0, then multiplying both sides of this equation by a#x7B;-1#x7D; yields b#x3D;0.",
"type": "quotation"
}
],
"glosses": [
"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."
],
"id": "en-zero_divisor-en-noun-lcgGPcI9",
"links": [
[
"algebra",
"algebra"
],
[
"ring",
"ring"
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],
"qualifier": "ring theory",
"raw_glosses": [
"(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."
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
}
],
"synonyms": [
{
"_dis1": "46 54",
"word": "zero-divisor"
}
],
"translations": [
{
"_dis1": "49 51",
"code": "fi",
"lang": "Finnish",
"sense": "element whose product with some nonzero element is zero",
"word": "nollanjakaja"
},
{
"_dis1": "49 51",
"code": "fr",
"lang": "French",
"sense": "element whose product with some nonzero element is zero",
"tags": [
"masculine"
],
"word": "diviseur de zéro"
}
],
"wikipedia": [
"zero divisor"
],
"word": "zero divisor"
}
{
"antonyms": [
{
"sense": "antonym(s) of \"any element whose product with some nonzero element is zero\"",
"word": "regular element"
}
],
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns"
],
"derived": [
{
"word": "zero divisor graph"
}
],
"forms": [
{
"form": "zero divisors",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "zero divisor (plural zero divisors)",
"name": "en-noun"
}
],
"hyponyms": [
{
"sense": "any element whose product with some nonzero element is zero",
"word": "trivial zero divisor"
},
{
"sense": "both senses",
"word": "exact zero divisor"
},
{
"sense": "both senses",
"word": "left zero divisor"
},
{
"sense": "both senses",
"word": "right zero divisor"
},
{
"sense": "both senses",
"word": "two-sided zero divisor"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "annihilator"
},
{
"word": "integral domain"
},
{
"word": "nilpotent"
}
],
"senses": [
{
"categories": [
"English terms with quotations",
"English terms with usage examples",
"en:Algebra"
],
"examples": [
{
"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"
},
{
"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": "quotation"
},
{
"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.",
"type": "quotation"
},
{
"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.",
"type": "quotation"
}
],
"glosses": [
"An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0."
],
"links": [
[
"algebra",
"algebra"
],
[
"ring",
"ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(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."
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"English terms with quotations",
"en:Algebra"
],
"examples": [
{
"ref": "2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234",
"text": "If R is an integral domain, that is, has no zero divisors, then R#x5B;x#x5D; also has no zero divisors.",
"type": "quotation"
},
{
"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.",
"type": "quotation"
},
{
"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 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#x3D;0, then multiplying both sides of this equation by a#x7B;-1#x7D; yields b#x3D;0.",
"type": "quotation"
}
],
"glosses": [
"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."
],
"links": [
[
"algebra",
"algebra"
],
[
"ring",
"ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(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."
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
}
],
"synonyms": [
{
"word": "zero-divisor"
}
],
"translations": [
{
"code": "fi",
"lang": "Finnish",
"sense": "element whose product with some nonzero element is zero",
"word": "nollanjakaja"
},
{
"code": "fr",
"lang": "French",
"sense": "element whose product with some nonzero element is zero",
"tags": [
"masculine"
],
"word": "diviseur de zéro"
}
],
"wikipedia": [
"zero divisor"
],
"word": "zero divisor"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-04-26 from the enwiktionary dump dated 2024-04-21 using wiktextract (93a6c53 and 21a9316). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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