See Noetherian ring on Wiktionary
{
"etymology_text": "Named after German mathematician Emmy Noether (1882–1935).",
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"derived": [
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"word": "left Noetherian ring"
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"word": "right Noetherian ring"
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"examples": [
{
"ref": "1986, Hideyuki Matsumura, translated by M. Reid, Commutative Ring Theory, Paperback edition, Cambridge University Press, published 1989, page ix:",
"text": "The central position occupied by Noetherian rings in commutative ring theory became evident from her^([Noether's]) work.",
"type": "quote"
},
{
"ref": "2000, John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian Rings, 2nd edition, American Mathematical Society, page 97:",
"text": "In this chapter the focus moves from semiprime rings to general Noetherian rings, although it does concentrate on prime and semiprime ideals.",
"type": "quote"
},
{
"text": "2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2nd Edition, page viii,\nDuring this same period, the explosive growth of the area of quantum groups provided a large new crop of noetherian rings to be analyzed, and thus gave major impetus to research in noetherian ring theory."
}
],
"glosses": [
"A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated)."
],
"hyponyms": [
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"sense": "Noetherian domain",
"word": "Dedekind domain"
},
{
"word": "Artinian ring"
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"id": "en-Noetherian_ring-en-noun-QnU228kn",
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"(algebra, ring theory) A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated)."
],
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"word": "Noetherian"
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"ipa": "/nə.ˈtɛ.ɹi.ən ˈɹɪŋɡ/"
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"word": "Noetherian ring"
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"word": "left Noetherian ring"
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"etymology_text": "Named after German mathematician Emmy Noether (1882–1935).",
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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"word": "left-Noetherian"
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"ref": "1986, Hideyuki Matsumura, translated by M. Reid, Commutative Ring Theory, Paperback edition, Cambridge University Press, published 1989, page ix:",
"text": "The central position occupied by Noetherian rings in commutative ring theory became evident from her^([Noether's]) work.",
"type": "quote"
},
{
"ref": "2000, John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian Rings, 2nd edition, American Mathematical Society, page 97:",
"text": "In this chapter the focus moves from semiprime rings to general Noetherian rings, although it does concentrate on prime and semiprime ideals.",
"type": "quote"
},
{
"text": "2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2nd Edition, page viii,\nDuring this same period, the explosive growth of the area of quantum groups provided a large new crop of noetherian rings to be analyzed, and thus gave major impetus to research in noetherian ring theory."
}
],
"glosses": [
"A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated)."
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"(algebra, ring theory) A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated)."
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"synonyms": [
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"word": "noetherian ring"
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],
"word": "Noetherian ring"
}
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