See integral domain in All languages combined, or Wiktionary
Download JSON data for integral domain meaning in English (4.9kB)
{
"forms": [
{
"form": "integral domains",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "integral domain (plural integral domains)",
"name": "en-noun"
}
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"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
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"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
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{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
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"examples": [
{
"text": "A ring R is an integral domain if and only if the polynomial ring R#x5B;x#x5D; is an integral domain.",
"type": "example"
},
{
"text": "For any integral domain there can be derived an associated field of fractions.",
"type": "example"
},
{
"ref": "1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266",
"text": "For integral domains, we will use a⁻¹ to designate the multiplicative inverse of a (if it has one; since not all elements need have inverses, this notation can be used only where it can be shown that an inverse exists).",
"type": "quotation"
},
{
"ref": "2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95",
"text": "An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product.",
"type": "quotation"
},
{
"ref": "2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171",
"text": "x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D; with p a prime, #x5B;#x5C;mathbb#x7B;Q#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;R#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, and #x5B;#x5C;mathbb#x7B;C#x7D;#x3B;#x2B;,#x5C;cdot#x5D; are all integral domains. The key example of an infinite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;. In fact, it is from #x5C;mathbb#x7B;Z#x7D; that the term integral domain is derived. Our main example of a finite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D;, when p is prime.",
"type": "quotation"
}
],
"glosses": [
"Any nonzero commutative ring in which the product of nonzero elements is nonzero."
],
"holonyms": [
{
"word": "field of fractions"
}
],
"hypernyms": [
{
"word": "commutative ring"
},
{
"word": "domain"
}
],
"hyponyms": [
{
"sense": "Euclidean domain",
"word": "field"
}
],
"id": "en-integral_domain-en-noun-OL98CWCr",
"links": [
[
"algebra",
"algebra"
],
[
"commutative ring",
"commutative ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero."
],
"synonyms": [
{
"sense": "commutative ring in which the product of nonzero elements is nonzero",
"word": "entire ring"
}
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"translations": [
{
"code": "fi",
"lang": "Finnish",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"word": "kokonaisalue"
},
{
"code": "fr",
"lang": "French",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "anneau d’intégrité"
},
{
"code": "it",
"lang": "Italian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "dominio d'integrità"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Cyrillic"
],
"word": "област цијелих"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Cyrillic"
],
"word": "област целих"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Cyrillic"
],
"word": "домен"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Latin"
],
"word": "oblast cijelih"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Latin"
],
"word": "oblast celih"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Latin"
],
"word": "domen"
},
{
"code": "es",
"lang": "Spanish",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "dominio de integridad"
}
],
"wikipedia": [
"integral domain"
]
}
],
"word": "integral domain"
}
{
"forms": [
{
"form": "integral domains",
"tags": [
"plural"
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}
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"expansion": "integral domain (plural integral domains)",
"name": "en-noun"
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"holonyms": [
{
"word": "field of fractions"
}
],
"hypernyms": [
{
"word": "commutative ring"
},
{
"word": "domain"
}
],
"hyponyms": [
{
"sense": "Euclidean domain",
"word": "field"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
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"English entries with incorrect language header",
"English lemmas",
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"English terms with quotations",
"English terms with usage examples",
"en:Algebra"
],
"examples": [
{
"text": "A ring R is an integral domain if and only if the polynomial ring R#x5B;x#x5D; is an integral domain.",
"type": "example"
},
{
"text": "For any integral domain there can be derived an associated field of fractions.",
"type": "example"
},
{
"ref": "1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266",
"text": "For integral domains, we will use a⁻¹ to designate the multiplicative inverse of a (if it has one; since not all elements need have inverses, this notation can be used only where it can be shown that an inverse exists).",
"type": "quotation"
},
{
"ref": "2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95",
"text": "An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product.",
"type": "quotation"
},
{
"ref": "2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171",
"text": "x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D; with p a prime, #x5B;#x5C;mathbb#x7B;Q#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;R#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, and #x5B;#x5C;mathbb#x7B;C#x7D;#x3B;#x2B;,#x5C;cdot#x5D; are all integral domains. The key example of an infinite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;. In fact, it is from #x5C;mathbb#x7B;Z#x7D; that the term integral domain is derived. Our main example of a finite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D;, when p is prime.",
"type": "quotation"
}
],
"glosses": [
"Any nonzero commutative ring in which the product of nonzero elements is nonzero."
],
"links": [
[
"algebra",
"algebra"
],
[
"commutative ring",
"commutative ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero."
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"integral domain"
]
}
],
"synonyms": [
{
"sense": "commutative ring in which the product of nonzero elements is nonzero",
"word": "entire ring"
}
],
"translations": [
{
"code": "fi",
"lang": "Finnish",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"word": "kokonaisalue"
},
{
"code": "fr",
"lang": "French",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "anneau d’intégrité"
},
{
"code": "it",
"lang": "Italian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "dominio d'integrità"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Cyrillic"
],
"word": "област цијелих"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Cyrillic"
],
"word": "област целих"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Cyrillic"
],
"word": "домен"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Latin"
],
"word": "oblast cijelih"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Latin"
],
"word": "oblast celih"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"Latin"
],
"word": "domen"
},
{
"code": "es",
"lang": "Spanish",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "dominio de integridad"
}
],
"word": "integral domain"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-03 from the enwiktionary dump dated 2024-05-02 using wiktextract (f4fd8c9 and c9440ce). 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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