See Dedekind domain in All languages combined, or Wiktionary
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{
"etymology_text": "Named after German mathematician Richard Dedekind (1831–1916).",
"forms": [
{
"form": "Dedekind domains",
"tags": [
"plural"
]
}
],
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"args": {},
"expansion": "Dedekind domain (plural Dedekind domains)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
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"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
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"derived": [
{
"word": "almost Dedekind domain"
}
],
"examples": [
{
"text": "It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible.",
"type": "example"
},
{
"ref": "1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Elsevier (Academic Press), page 201",
"text": "In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.",
"type": "quotation"
},
{
"ref": "2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55",
"text": "As we can see every principal ideal domain is a Dedekind domain.",
"type": "quotation"
},
{
"ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 266",
"text": "Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals.",
"type": "quotation"
}
],
"glosses": [
"An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations)."
],
"hypernyms": [
{
"sense": "integral domain whose prime ideals factorise uniquely",
"word": "Noetherian domain"
}
],
"id": "en-Dedekind_domain-en-noun-yP-R08SM",
"links": [
[
"algebra",
"algebra"
],
[
"ring theory",
"ring theory"
],
[
"integral domain",
"integral domain"
],
[
"ideal",
"ideal"
],
[
"prime ideal",
"prime ideal"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations)."
],
"synonyms": [
{
"sense": "integral domain whose prime ideals factorise uniquely",
"word": "Dedekind ring"
}
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "anneau de Dedekind"
},
{
"code": "de",
"lang": "German",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "Dedekindring"
},
{
"code": "it",
"lang": "Italian",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "dominio di Dedekind"
},
{
"code": "it",
"lang": "Italian",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "anello di Dedekind"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "domínio de Dedekind"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "anel de Dedekind"
}
],
"wikipedia": [
"Dedekind domain",
"Richard Dedekind"
]
}
],
"word": "Dedekind domain"
}
{
"derived": [
{
"word": "almost Dedekind domain"
}
],
"etymology_text": "Named after German mathematician Richard Dedekind (1831–1916).",
"forms": [
{
"form": "Dedekind domains",
"tags": [
"plural"
]
}
],
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"expansion": "Dedekind domain (plural Dedekind domains)",
"name": "en-noun"
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],
"hypernyms": [
{
"sense": "integral domain whose prime ideals factorise uniquely",
"word": "Noetherian domain"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
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],
"examples": [
{
"text": "It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible.",
"type": "example"
},
{
"ref": "1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Elsevier (Academic Press), page 201",
"text": "In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.",
"type": "quotation"
},
{
"ref": "2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55",
"text": "As we can see every principal ideal domain is a Dedekind domain.",
"type": "quotation"
},
{
"ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 266",
"text": "Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals.",
"type": "quotation"
}
],
"glosses": [
"An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations)."
],
"links": [
[
"algebra",
"algebra"
],
[
"ring theory",
"ring theory"
],
[
"integral domain",
"integral domain"
],
[
"ideal",
"ideal"
],
[
"prime ideal",
"prime ideal"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations)."
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"Dedekind domain",
"Richard Dedekind"
]
}
],
"synonyms": [
{
"sense": "integral domain whose prime ideals factorise uniquely",
"word": "Dedekind ring"
}
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "anneau de Dedekind"
},
{
"code": "de",
"lang": "German",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "Dedekindring"
},
{
"code": "it",
"lang": "Italian",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "dominio di Dedekind"
},
{
"code": "it",
"lang": "Italian",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "anello di Dedekind"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "domínio de Dedekind"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "integral domain whose prime ideals factorise uniquely",
"tags": [
"masculine"
],
"word": "anel de Dedekind"
}
],
"word": "Dedekind 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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