See Cohen-Macaulay in All languages combined, or Wiktionary
Download JSON data for Cohen-Macaulay meaning in English (4.5kB)
{
"etymology_text": "Named for Irvin Cohen and Francis Sowerby Macaulay, who proved unmixedness results for specific classes of rings, which Cohen-Macaulay rings generalize.",
"head_templates": [
{
"args": {
"1": "-"
},
"expansion": "Cohen-Macaulay (not comparable)",
"name": "en-adj"
}
],
"lang": "English",
"lang_code": "en",
"pos": "adj",
"senses": [
{
"categories": [
{
"_dis": "35 13 39 13",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "30 20 30 20",
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w+disamb"
}
],
"glosses": [
"Such that its depth is equal to its Krull dimension."
],
"id": "en-Cohen-Macaulay-en-adj-en:module",
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"finite",
"finite"
],
[
"module",
"module"
],
[
"noetherian",
"noetherian"
],
[
"local",
"local#English:_ring"
],
[
"ring",
"ring"
],
[
"depth",
"depth"
],
[
"Krull dimension",
"Krull dimension"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a finite module over a noetherian local ring) Such that its depth is equal to its Krull dimension."
],
"raw_tags": [
"of a finite module over a noetherian local ring"
],
"senseid": [
"en:module"
],
"tags": [
"not-comparable"
]
},
{
"categories": [
{
"_dis": "35 13 39 13",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "30 20 30 20",
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w+disamb"
}
],
"glosses": [
"Cohen-Macaulay as a module over itself."
],
"id": "en-Cohen-Macaulay-en-adj-sV0Mz~Gk",
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"noetherian",
"noetherian"
],
[
"local",
"local"
],
[
"ring",
"ring"
],
[
"Cohen-Macaulay",
"Cohen-Macaulay#English:_module"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a noetherian local ring) Cohen-Macaulay as a module over itself."
],
"raw_tags": [
"of a noetherian local ring"
],
"tags": [
"not-comparable"
]
},
{
"categories": [
{
"_dis": "35 13 39 13",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "30 20 30 20",
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w+disamb"
}
],
"glosses": [
"Such that all localizations of M at maximal ideals contained in the support of M are either Cohen-Macaulay or trivial."
],
"id": "en-Cohen-Macaulay-en-adj-en:module2",
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"module",
"module"
],
[
"noetherian",
"noetherian"
],
[
"ring",
"ring"
],
[
"localization",
"localization"
],
[
"maximal ideal",
"maximal ideal"
],
[
"support",
"support"
],
[
"Cohen-Macaulay",
"Cohen-Macaulay#English:_module"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a module M over a noetherian ring) Such that all localizations of M at maximal ideals contained in the support of M are either Cohen-Macaulay or trivial."
],
"raw_tags": [
"of a module M over a noetherian ring"
],
"senseid": [
"en:module2"
],
"tags": [
"not-comparable"
]
},
{
"categories": [
{
"_dis": "35 13 39 13",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "30 20 30 20",
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w+disamb"
}
],
"glosses": [
"Cohen-Macaulay as a module over itself."
],
"id": "en-Cohen-Macaulay-en-adj-sV0Mz~Gk1",
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"noetherian",
"noetherian"
],
[
"ring",
"ring"
],
[
"Cohen-Macaulay",
"Cohen-Macaulay#English:_module2"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a noetherian ring) Cohen-Macaulay as a module over itself."
],
"raw_tags": [
"of a noetherian ring"
],
"tags": [
"not-comparable"
]
}
],
"wikipedia": [
"Francis Sowerby Macaulay",
"Irvin Cohen"
],
"word": "Cohen-Macaulay"
}
{
"categories": [
"English adjectives",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English uncomparable adjectives",
"en:Algebra"
],
"etymology_text": "Named for Irvin Cohen and Francis Sowerby Macaulay, who proved unmixedness results for specific classes of rings, which Cohen-Macaulay rings generalize.",
"head_templates": [
{
"args": {
"1": "-"
},
"expansion": "Cohen-Macaulay (not comparable)",
"name": "en-adj"
}
],
"lang": "English",
"lang_code": "en",
"pos": "adj",
"senses": [
{
"glosses": [
"Such that its depth is equal to its Krull dimension."
],
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"finite",
"finite"
],
[
"module",
"module"
],
[
"noetherian",
"noetherian"
],
[
"local",
"local#English:_ring"
],
[
"ring",
"ring"
],
[
"depth",
"depth"
],
[
"Krull dimension",
"Krull dimension"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a finite module over a noetherian local ring) Such that its depth is equal to its Krull dimension."
],
"raw_tags": [
"of a finite module over a noetherian local ring"
],
"senseid": [
"en:module"
],
"tags": [
"not-comparable"
]
},
{
"glosses": [
"Cohen-Macaulay as a module over itself."
],
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"noetherian",
"noetherian"
],
[
"local",
"local"
],
[
"ring",
"ring"
],
[
"Cohen-Macaulay",
"Cohen-Macaulay#English:_module"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a noetherian local ring) Cohen-Macaulay as a module over itself."
],
"raw_tags": [
"of a noetherian local ring"
],
"tags": [
"not-comparable"
]
},
{
"glosses": [
"Such that all localizations of M at maximal ideals contained in the support of M are either Cohen-Macaulay or trivial."
],
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"module",
"module"
],
[
"noetherian",
"noetherian"
],
[
"ring",
"ring"
],
[
"localization",
"localization"
],
[
"maximal ideal",
"maximal ideal"
],
[
"support",
"support"
],
[
"Cohen-Macaulay",
"Cohen-Macaulay#English:_module"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a module M over a noetherian ring) Such that all localizations of M at maximal ideals contained in the support of M are either Cohen-Macaulay or trivial."
],
"raw_tags": [
"of a module M over a noetherian ring"
],
"senseid": [
"en:module2"
],
"tags": [
"not-comparable"
]
},
{
"glosses": [
"Cohen-Macaulay as a module over itself."
],
"links": [
[
"commutative algebra",
"commutative algebra"
],
[
"noetherian",
"noetherian"
],
[
"ring",
"ring"
],
[
"Cohen-Macaulay",
"Cohen-Macaulay#English:_module2"
]
],
"qualifier": "commutative algebra",
"raw_glosses": [
"(commutative algebra, of a noetherian ring) Cohen-Macaulay as a module over itself."
],
"raw_tags": [
"of a noetherian ring"
],
"tags": [
"not-comparable"
]
}
],
"wikipedia": [
"Francis Sowerby Macaulay",
"Irvin Cohen"
],
"word": "Cohen-Macaulay"
}
This page is a part of the kaikki.org machine-readable English 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.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.