. . . . . . . . "Maticov\u00FD okruh"@cs . . . . . "1097451145"^^ . . . "\u77E9\u9635\u73AF\u5C31\u662F\u8003\u616E\u77E9\u9635\u5728\u74B0R\u4E0B\u7D93\u7531\u77E9\u9635\u52A0\u6CD5\u548C\u77E9\u9635\u4E58\u6CD5\u5F62\u6210\u7684\u73AF\uFF0C\u5F9E\u74B0R\u4E2D\u7684\u5143\u7D20\u7D44\u6210\u7684n\u00D7n \u65B9\u9635\u5F62\u6210\u7684\u77E9\u9663\u74B0\u8A18\u4F5CMn(R)\uFF0C\u67D0\u4E9B\u65E0\u9650\u9636\u77E9\u9635\u4E5F\u53EF\u4EE5\u7D44\u6210\u65E0\u9650\u77E9\u9635\u73AF\uFF0C\u4EFB\u4F55\u77E9\u9635\u73AF\u7684\u5B50\u73AF\u4E5F\u90FD\u662F\u77E9\u9663\u73AF\u3002\u5982 R\u200B\u200B\u662F\u4E00\u4E2A\u4EA4\u6362\u73AF\uFF0C\u5219\u77E9\u9635\u73AFMn(R)\u662F\u4E00\u4E2A\u7ED3\u5408\u4EE3\u6570\uFF0C\u88AB\u79F0\u4E3A\u3002\u5728\u8FD9\u79CD\u60C5\u51B5\u4E0B\uFF0C\u5982\u679C M\u662F\u4E00\u4E2A\u77E9\u9635\uFF0C r\u2208 R\uFF0C\u90A3\u4E48\u77E9\u9635Mr\u4E5F\u662F\u77E9\u9635\uFF0C\u5176\u77E9\u9663\u5143\u70BAM\u7684\u77E9\u9663\u5143\u4E58r\u3002 \u9019\u7BC7\u6587\u7AE0\u5047\u8BBER\u662F\u53EF\u7D50\u5408\u74B0\u4E14\u5355\u4F4D1\u22600\uFF08\u5355\u4F4D1=0\u7684\u53EA\u6709\u96F6\u73AF)\uFF0C\u867D\u7136\u6CA1\u6709\u5355\u4F4D\u4E5F\u53EF\u4EE5\u5F62\u6210\u77E9\u9663\u74B0\u3002"@zh . . . . "In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n \u00D7 n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn\u00D7n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs."@en . "Matrix ring"@en . . . . . . . "12941"^^ . . . "In de abstracte algebra, een deelgebied van de wiskunde, is de matrixring de verzameling van alle -matrices over een willekeurige ring . Deze verzameling is zelf een ring onder de operaties matrixoptelling en matrixvermenigvuldiging."@nl . . . . . . . . . . . . . . "\u884C\u5217\u74B0"@ja . . . "\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u884C\u5217\u74B0 (matrix ring) \u306F\u3001\u304A\u3088\u3073\u884C\u5217\u306E\u4E57\u6CD5\u306E\u3082\u3068\u3067\u74B0\u3092\u306A\u3059\u3001\u884C\u5217\u306E\u4EFB\u610F\u306E\u96C6\u307E\u308A\u3067\u3042\u308B\u3002\u5225\u306E\u74B0\u3092\u6210\u5206\u306B\u6301\u3064 n\u00D7n \u884C\u5217\u5168\u4F53\u306E\u96C6\u5408\u3084\u7121\u9650\u6B21\u884C\u5217\u74B0 (infinite matrix ring) \u3092\u306A\u3059\u7121\u9650\u6B21\u884C\u5217\u306E\u3042\u308B\u90E8\u5206\u96C6\u5408\u306F\u884C\u5217\u74B0\u3067\u3042\u308B\u3002\u3053\u308C\u3089\u306E\u884C\u5217\u74B0\u306E\u4EFB\u610F\u306E\u90E8\u5206\u74B0\u3082\u307E\u305F\u884C\u5217\u74B0\u3067\u3042\u308B\u3002 R \u304C\u53EF\u63DB\u74B0\u306E\u3068\u304D\u3001\u884C\u5217\u74B0 Mn(R) \u306F\u884C\u5217\u591A\u5143\u74B0 (matrix algebra) \u3068\u547C\u3070\u308C\u308B\u7D50\u5408\u591A\u5143\u74B0\u3067\u3042\u308B\u3002\u3053\u306E\u72B6\u6CC1\u306B\u304A\u3044\u3066\u3001M \u304C\u884C\u5217\u3067 r \u304C R \u306E\u5143\u3067\u3042\u308C\u3070\u3001\u884C\u5217 Mr \u306F\u884C\u5217 M \u306E\u5404\u6210\u5206\u306B r \u3092\u304B\u3051\u305F\u3082\u306E\u3067\u3042\u308B\u3002 \u884C\u5217\u74B0\u306F\u5358\u4F4D\u5143\u3092\u3082\u305F\u306A\u3044\u74B0\u4E0A\u4F5C\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u304C\u3001\u7D42\u59CB R \u306F\u5358\u4F4D\u5143 1 \u2260 0 \u3092\u3082\u3064\u7D50\u5408\u7684\u74B0\u3067\u3042\u308B\u3068\u4EEE\u5B9A\u3059\u308B\u3002"@ja . . . . "Der Matrizenring, Matrixring oder Ring der Matrizen ist in der Mathematik der Ring der quadratischen Matrizen fester Gr\u00F6\u00DFe mit Eintr\u00E4gen aus einem weiteren, zugrunde liegenden Ring. Die additive und die multiplikative Verkn\u00FCpfung im Matrizenring sind die Matrizenaddition und die Matrizenmultiplikation. Das neutrale Element im Matrizenring ist die Nullmatrix und das Einselement die Einheitsmatrix. Der Matrizenring ist zu seinem zugrunde liegenden Ring und erbt daher viele seiner Eigenschaften. Allerdings ist der Matrizenring im Allgemeinen nicht kommutativ, selbst wenn der zugrunde liegende Ring kommutativ sein sollte. Der Matrizenring besitzt in der Ringtheorie eine besondere Bedeutung, da jeder Endomorphismenring eines freien Moduls mit endlicher Basis isomorph zu einem Matrizenring ist. Viele Ringe lassen sich somit als Unterring eines Matrizenrings realisieren. Dieses Vorgehen nennt man in Analogie zur Permutationsdarstellung einer Gruppe Matrixdarstellung des Rings."@de . "\u77E9\u9635\u73AF"@zh . . . . "\u77E9\u9635\u73AF\u5C31\u662F\u8003\u616E\u77E9\u9635\u5728\u74B0R\u4E0B\u7D93\u7531\u77E9\u9635\u52A0\u6CD5\u548C\u77E9\u9635\u4E58\u6CD5\u5F62\u6210\u7684\u73AF\uFF0C\u5F9E\u74B0R\u4E2D\u7684\u5143\u7D20\u7D44\u6210\u7684n\u00D7n \u65B9\u9635\u5F62\u6210\u7684\u77E9\u9663\u74B0\u8A18\u4F5CMn(R)\uFF0C\u67D0\u4E9B\u65E0\u9650\u9636\u77E9\u9635\u4E5F\u53EF\u4EE5\u7D44\u6210\u65E0\u9650\u77E9\u9635\u73AF\uFF0C\u4EFB\u4F55\u77E9\u9635\u73AF\u7684\u5B50\u73AF\u4E5F\u90FD\u662F\u77E9\u9663\u73AF\u3002\u5982 R\u200B\u200B\u662F\u4E00\u4E2A\u4EA4\u6362\u73AF\uFF0C\u5219\u77E9\u9635\u73AFMn(R)\u662F\u4E00\u4E2A\u7ED3\u5408\u4EE3\u6570\uFF0C\u88AB\u79F0\u4E3A\u3002\u5728\u8FD9\u79CD\u60C5\u51B5\u4E0B\uFF0C\u5982\u679C M\u662F\u4E00\u4E2A\u77E9\u9635\uFF0C r\u2208 R\uFF0C\u90A3\u4E48\u77E9\u9635Mr\u4E5F\u662F\u77E9\u9635\uFF0C\u5176\u77E9\u9663\u5143\u70BAM\u7684\u77E9\u9663\u5143\u4E58r\u3002 \u9019\u7BC7\u6587\u7AE0\u5047\u8BBER\u662F\u53EF\u7D50\u5408\u74B0\u4E14\u5355\u4F4D1\u22600\uFF08\u5355\u4F4D1=0\u7684\u53EA\u6709\u96F6\u73AF)\uFF0C\u867D\u7136\u6CA1\u6709\u5355\u4F4D\u4E5F\u53EF\u4EE5\u5F62\u6210\u77E9\u9663\u74B0\u3002"@zh . . . . . "487541"^^ . . . . . . . . . . . "In de abstracte algebra, een deelgebied van de wiskunde, is de matrixring de verzameling van alle -matrices over een willekeurige ring . Deze verzameling is zelf een ring onder de operaties matrixoptelling en matrixvermenigvuldiging."@nl . . "Maticov\u00FD okruh je pojem abstraktn\u00ED algebry. Maticov\u00FDm okruhem M(n,R) se rozum\u00ED mno\u017Eina tvo\u0159en\u00E1 \u010Dtvercov\u00FDmi maticemi \u0159\u00E1du n s prvky z okruhu R, kter\u00E1 spolu se standardn\u00EDmi operacemi s\u010D\u00EDt\u00E1n\u00ED matic a n\u00E1soben\u00ED matic tvo\u0159\u00ED okruh."@cs . . . . . . . . . . . . . . . . . . . . "\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u884C\u5217\u74B0 (matrix ring) \u306F\u3001\u304A\u3088\u3073\u884C\u5217\u306E\u4E57\u6CD5\u306E\u3082\u3068\u3067\u74B0\u3092\u306A\u3059\u3001\u884C\u5217\u306E\u4EFB\u610F\u306E\u96C6\u307E\u308A\u3067\u3042\u308B\u3002\u5225\u306E\u74B0\u3092\u6210\u5206\u306B\u6301\u3064 n\u00D7n \u884C\u5217\u5168\u4F53\u306E\u96C6\u5408\u3084\u7121\u9650\u6B21\u884C\u5217\u74B0 (infinite matrix ring) \u3092\u306A\u3059\u7121\u9650\u6B21\u884C\u5217\u306E\u3042\u308B\u90E8\u5206\u96C6\u5408\u306F\u884C\u5217\u74B0\u3067\u3042\u308B\u3002\u3053\u308C\u3089\u306E\u884C\u5217\u74B0\u306E\u4EFB\u610F\u306E\u90E8\u5206\u74B0\u3082\u307E\u305F\u884C\u5217\u74B0\u3067\u3042\u308B\u3002 R \u304C\u53EF\u63DB\u74B0\u306E\u3068\u304D\u3001\u884C\u5217\u74B0 Mn(R) \u306F\u884C\u5217\u591A\u5143\u74B0 (matrix algebra) \u3068\u547C\u3070\u308C\u308B\u7D50\u5408\u591A\u5143\u74B0\u3067\u3042\u308B\u3002\u3053\u306E\u72B6\u6CC1\u306B\u304A\u3044\u3066\u3001M \u304C\u884C\u5217\u3067 r \u304C R \u306E\u5143\u3067\u3042\u308C\u3070\u3001\u884C\u5217 Mr \u306F\u884C\u5217 M \u306E\u5404\u6210\u5206\u306B r \u3092\u304B\u3051\u305F\u3082\u306E\u3067\u3042\u308B\u3002 \u884C\u5217\u74B0\u306F\u5358\u4F4D\u5143\u3092\u3082\u305F\u306A\u3044\u74B0\u4E0A\u4F5C\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u304C\u3001\u7D42\u59CB R \u306F\u5358\u4F4D\u5143 1 \u2260 0 \u3092\u3082\u3064\u7D50\u5408\u7684\u74B0\u3067\u3042\u308B\u3068\u4EEE\u5B9A\u3059\u308B\u3002"@ja . . . . . "Matrizenring"@de . . . . . . "Maticov\u00FD okruh je pojem abstraktn\u00ED algebry. Maticov\u00FDm okruhem M(n,R) se rozum\u00ED mno\u017Eina tvo\u0159en\u00E1 \u010Dtvercov\u00FDmi maticemi \u0159\u00E1du n s prvky z okruhu R, kter\u00E1 spolu se standardn\u00EDmi operacemi s\u010D\u00EDt\u00E1n\u00ED matic a n\u00E1soben\u00ED matic tvo\u0159\u00ED okruh."@cs . "In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n \u00D7 n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn\u00D7n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring Mn(R) is an associative algebra over R, and may be called a matrix algebra. In this setting, if M is a matrix and r is in R, then the matrix rM is the matrix M with each of its entries multiplied by r."@en . . . . "Der Matrizenring, Matrixring oder Ring der Matrizen ist in der Mathematik der Ring der quadratischen Matrizen fester Gr\u00F6\u00DFe mit Eintr\u00E4gen aus einem weiteren, zugrunde liegenden Ring. Die additive und die multiplikative Verkn\u00FCpfung im Matrizenring sind die Matrizenaddition und die Matrizenmultiplikation. Das neutrale Element im Matrizenring ist die Nullmatrix und das Einselement die Einheitsmatrix. Der Matrizenring ist zu seinem zugrunde liegenden Ring und erbt daher viele seiner Eigenschaften. Allerdings ist der Matrizenring im Allgemeinen nicht kommutativ, selbst wenn der zugrunde liegende Ring kommutativ sein sollte."@de . . . . . "Matrixring"@nl . . . . . . . . . . . . .