See central simple algebra in All languages combined, or Wiktionary
{ "forms": [ { "form": "central simple algebras", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "central simple algebra (plural central simple algebras)", "name": "en-noun" } ], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [ { "categories": [ { "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w" }, { "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w" }, { "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w" }, { "kind": "other", "name": "Pages with entries", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Dutch translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Portuguese translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Spanish translations", "parents": [], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Algebra", "orig": "en:Algebra", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "examples": [ { "text": "The complex numbers #92;C form a central simple algebra over themselves, but not over the real numbers #92;R (the centre of #92;C is all of #92;C, not just #92;R). The quaternions #92;mathbbH form a 4-dimensional central simple algebra over #92;R.", "type": "example" }, { "text": "The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).", "type": "example" }, { "ref": "1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 151:", "text": "This crossed product E#92;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.", "type": "quote" }, { "ref": "2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:", "text": "Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra D.", "type": "quote" }, { "ref": "2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:", "text": "Let A#95;1,A#95;2 be central simple algebras over a field K. Then A#95;1#92;otimes#95;KA#95;2 can be shown to be a central simple algebra over K. Further, if A is a central simple algebra over a field K, then A#92;otimes#95;KA#92;operatorname#123;op#125;#92;cong#92;operatorname#123;Aut#125;#95;#123;K#92;operatorname#123;-Vect", "type": "quote" } ], "glosses": [ "A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K." ], "hypernyms": [ { "word": "simple algebra" } ], "id": "en-central_simple_algebra-en-noun-vHMwRn4z", "links": [ [ "algebra", "algebra" ], [ "associative algebra", "associative algebra#English" ], [ "field", "field#English:_in_mathematics" ], [ "simple algebra", "simple algebra#English" ], [ "centre", "center#English" ] ], "qualifier": "ring theory", "raw_glosses": [ "(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K." ], "synonyms": [ { "tags": [ "initialism" ], "word": "CSA" } ], "topics": [ "algebra", "mathematics", "sciences" ], "translations": [ { "code": "nl", "lang": "Dutch", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "centrale enkelvoudige algebra" }, { "code": "nl", "lang": "Dutch", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "CEA" }, { "code": "pt", "lang": "Portuguese", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "álgebra simple central" }, { "code": "pt", "lang": "Portuguese", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "ASC" }, { "code": "es", "lang": "Spanish", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "álgebra simple central" }, { "code": "es", "lang": "Spanish", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "ASC" } ], "wikipedia": [ "central simple algebra" ] } ], "word": "central simple algebra" }
{ "forms": [ { "form": "central simple algebras", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "central simple algebra (plural central simple algebras)", "name": "en-noun" } ], "hypernyms": [ { "word": "simple algebra" } ], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [ { "categories": [ "English countable nouns", "English entries with incorrect language header", "English lemmas", "English multiword terms", "English nouns", "English terms with quotations", "English terms with usage examples", "Entries with translation boxes", "Pages with 1 entry", "Pages with entries", "Terms with Dutch translations", "Terms with Portuguese translations", "Terms with Spanish translations", "en:Algebra" ], "examples": [ { "text": "The complex numbers #92;C form a central simple algebra over themselves, but not over the real numbers #92;R (the centre of #92;C is all of #92;C, not just #92;R). The quaternions #92;mathbbH form a 4-dimensional central simple algebra over #92;R.", "type": "example" }, { "text": "The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).", "type": "example" }, { "ref": "1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 151:", "text": "This crossed product E#92;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.", "type": "quote" }, { "ref": "2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:", "text": "Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra D.", "type": "quote" }, { "ref": "2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:", "text": "Let A#95;1,A#95;2 be central simple algebras over a field K. Then A#95;1#92;otimes#95;KA#95;2 can be shown to be a central simple algebra over K. Further, if A is a central simple algebra over a field K, then A#92;otimes#95;KA#92;operatorname#123;op#125;#92;cong#92;operatorname#123;Aut#125;#95;#123;K#92;operatorname#123;-Vect", "type": "quote" } ], "glosses": [ "A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K." ], "links": [ [ "algebra", "algebra" ], [ "associative algebra", "associative algebra#English" ], [ "field", "field#English:_in_mathematics" ], [ "simple algebra", "simple algebra#English" ], [ "centre", "center#English" ] ], "qualifier": "ring theory", "raw_glosses": [ "(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K." ], "topics": [ "algebra", "mathematics", "sciences" ], "wikipedia": [ "central simple algebra" ] } ], "synonyms": [ { "tags": [ "initialism" ], "word": "CSA" } ], "translations": [ { "code": "nl", "lang": "Dutch", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "centrale enkelvoudige algebra" }, { "code": "nl", "lang": "Dutch", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "CEA" }, { "code": "pt", "lang": "Portuguese", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "álgebra simple central" }, { "code": "pt", "lang": "Portuguese", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "ASC" }, { "code": "es", "lang": "Spanish", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "álgebra simple central" }, { "code": "es", "lang": "Spanish", "sense": "type of associative algebra over a field", "tags": [ "feminine" ], "word": "ASC" } ], "word": "central simple algebra" }
Download raw JSONL data for central simple algebra meaning in English (4.0kB)
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