See central simple algebra in All languages combined, or Wiktionary
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"text": "The complex numbers #x5C;C form a central simple algebra over themselves, but not over the real numbers #x5C;R (the centre of #x5C;C is all of #x5C;C, not just #x5C;R). The quaternions #x5C;mathbbH form a 4-dimensional central simple algebra over #x5C;R.",
"type": "example"
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"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"
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{
"ref": "1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 151",
"text": "This crossed product E#x5C;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.",
"type": "quotation"
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{
"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": "quotation"
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{
"ref": "2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84",
"text": "Let A#x5F;1,A#x5F;2 be central simple algebras over a field K. Then A#x5F;1#x5C;otimes#x5F;KA#x5F;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#x5C;otimes#x5F;KA#x5C;operatorname#x7B;op#x7D;#x5C;cong#x5C;operatorname#x7B;Aut#x7D;#x5F;#x7B;K#x5C;operatorname#x7B;-Vect(A). I.e., it is isomorphic to a matrix algebra.}}",
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"A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K."
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"(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K."
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"word": "álgebra simple central"
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"word": "álgebra simple central"
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"text": "The complex numbers #x5C;C form a central simple algebra over themselves, but not over the real numbers #x5C;R (the centre of #x5C;C is all of #x5C;C, not just #x5C;R). The quaternions #x5C;mathbbH form a 4-dimensional central simple algebra over #x5C;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#x5C;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.",
"type": "quotation"
},
{
"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": "quotation"
},
{
"ref": "2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84",
"text": "Let A#x5F;1,A#x5F;2 be central simple algebras over a field K. Then A#x5F;1#x5C;otimes#x5F;KA#x5F;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#x5C;otimes#x5F;KA#x5C;operatorname#x7B;op#x7D;#x5C;cong#x5C;operatorname#x7B;Aut#x7D;#x5F;#x7B;K#x5C;operatorname#x7B;-Vect(A). I.e., it is isomorphic to a matrix algebra.}}",
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"(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K."
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"sense": "type of associative algebra over a field",
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"word": "centrale enkelvoudige algebra"
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