Matrix ring (original) (raw)
Maticový okruh je pojem abstraktní algebry. Maticovým okruhem M(n,R) se rozumí množina tvořená čtvercovými maticemi řádu n s prvky z okruhu R, která spolu se standardními operacemi sčítání matic a násobení matic tvoří okruh.
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| dbo:abstract | Maticový okruh je pojem abstraktní algebry. Maticovým okruhem M(n,R) se rozumí množina tvořená čtvercovými maticemi řádu n s prvky z okruhu R, která spolu se standardními operacemi sčítání matic a násobení matic tvoří okruh. (cs) Der Matrizenring, Matrixring oder Ring der Matrizen ist in der Mathematik der Ring der quadratischen Matrizen fester Größe mit Einträgen aus einem weiteren, zugrunde liegenden Ring. Die additive und die multiplikative Verknüpfung 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) 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 × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). 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) 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) 抽象代数学において、行列環 (matrix ring) は、および行列の乗法のもとで環をなす、行列の任意の集まりである。別の環を成分に持つ n×n 行列全体の集合や無限次行列環 (infinite matrix ring) をなす無限次行列のある部分集合は行列環である。これらの行列環の任意の部分環もまた行列環である。 R が可換環のとき、行列環 Mn(R) は行列多元環 (matrix algebra) と呼ばれる結合多元環である。この状況において、M が行列で r が R の元であれば、行列 Mr は行列 M の各成分に r をかけたものである。 行列環は単位元をもたない環上作ることができるが、終始 R は単位元 1 ≠ 0 をもつ結合的環であると仮定する。 (ja) 矩阵环就是考慮矩阵在環R下經由矩阵加法和矩阵乘法形成的环，從環R中的元素組成的n×n 方阵形成的矩陣環記作Mn(R)，某些无限阶矩阵也可以組成无限矩阵环，任何矩阵环的子环也都是矩陣环。如 R是一个交换环，则矩阵环Mn(R)是一个结合代数，被称为。在这种情况下，如果 M是一个矩阵， r∈ R，那么矩阵Mr也是矩阵，其矩陣元為M的矩陣元乘r。 這篇文章假设R是可結合環且单位1≠0（单位1=0的只有零环)，虽然没有单位也可以形成矩陣環。 (zh) |
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| rdfs:comment | Maticový okruh je pojem abstraktní algebry. Maticovým okruhem M(n,R) se rozumí množina tvořená čtvercovými maticemi řádu n s prvky z okruhu R, která spolu se standardními operacemi sčítání matic a násobení matic tvoří okruh. (cs) 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) 抽象代数学において、行列環 (matrix ring) は、および行列の乗法のもとで環をなす、行列の任意の集まりである。別の環を成分に持つ n×n 行列全体の集合や無限次行列環 (infinite matrix ring) をなす無限次行列のある部分集合は行列環である。これらの行列環の任意の部分環もまた行列環である。 R が可換環のとき、行列環 Mn(R) は行列多元環 (matrix algebra) と呼ばれる結合多元環である。この状況において、M が行列で r が R の元であれば、行列 Mr は行列 M の各成分に r をかけたものである。 行列環は単位元をもたない環上作ることができるが、終始 R は単位元 1 ≠ 0 をもつ結合的環であると仮定する。 (ja) 矩阵环就是考慮矩阵在環R下經由矩阵加法和矩阵乘法形成的环，從環R中的元素組成的n×n 方阵形成的矩陣環記作Mn(R)，某些无限阶矩阵也可以組成无限矩阵环，任何矩阵环的子环也都是矩陣环。如 R是一个交换环，则矩阵环Mn(R)是一个结合代数，被称为。在这种情况下，如果 M是一个矩阵， r∈ R，那么矩阵Mr也是矩阵，其矩陣元為M的矩陣元乘r。 這篇文章假设R是可結合環且单位1≠0（单位1=0的只有零环)，虽然没有单位也可以形成矩陣環。 (zh) Der Matrizenring, Matrixring oder Ring der Matrizen ist in der Mathematik der Ring der quadratischen Matrizen fester Größe mit Einträgen aus einem weiteren, zugrunde liegenden Ring. Die additive und die multiplikative Verknüpfung 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) 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 × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). 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) |
| rdfs:label | Maticový okruh (cs) Matrizenring (de) 行列環 (ja) Matrix ring (en) Matrixring (nl) 矩阵环 (zh) |
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