Free algebra (original) (raw)
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra.
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| dbo:abstract | In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. (en) En álgebra abstracta, el álgebra libre es el análogo no conmutativo del anillo de polinomios. Sea R un anillo. El álgebra libre en n indeterminadas, X1..., Xn, es el anillo generado por todas las combinaciones lineales de los productos de las variables. Este anillo es denotado por R<X1..., Xn>. A diferencia de un anillo polinómico, las variables no conmutan. Por ejemplo X1X2 no es igual a X2X1. Sobre un cuerpo, el álgebra libre en n indeterminadas se puede construir como el álgebra tensorial de un espacio vectorial n-dimensional. (Para un anillo de coeficientes más general, la misma construcción funciona si tomamos el módulo libre en n generadores.) * Datos: Q5500180 (es) En mathématiques, un polynôme non commutatif est une combinaison linéaire de mots sur des indéterminées non commutatives. On distingue ainsi les monômes x2y, xyx et yx2. L’ensemble des polynômes non commutatifs sur des variables x1, x2, … et à coefficients dans un anneau R est l’algèbre associative libre notée R⟨x1, x2, …⟩. Les polynômes non commutatifs peuvent définir des identités polynomiales sur des algèbres associatives. * Portail de l’algèbre (fr) 数学、とくに環論という抽象代数学の分野において、自由代数（じゆうだいすう、英: free algebra）は多項式環の非可換類似である、なぜならばその元は可換でない変数の「多項式」として書けるからである。同様に、多項式環は自由可換代数 (free commutative algebra) と見ることができる（多項式環#多項式環の普遍性参照）。 (ja) |
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| dbp:author | L.A. Bokut' (en) |
| dbp:id | f/f041520 (en) |
| dbp:title | Free associative algebra (en) |
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| rdfs:comment | In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. (en) En mathématiques, un polynôme non commutatif est une combinaison linéaire de mots sur des indéterminées non commutatives. On distingue ainsi les monômes x2y, xyx et yx2. L’ensemble des polynômes non commutatifs sur des variables x1, x2, … et à coefficients dans un anneau R est l’algèbre associative libre notée R⟨x1, x2, …⟩. Les polynômes non commutatifs peuvent définir des identités polynomiales sur des algèbres associatives. * Portail de l’algèbre (fr) 数学、とくに環論という抽象代数学の分野において、自由代数（じゆうだいすう、英: free algebra）は多項式環の非可換類似である、なぜならばその元は可換でない変数の「多項式」として書けるからである。同様に、多項式環は自由可換代数 (free commutative algebra) と見ることができる（多項式環#多項式環の普遍性参照）。 (ja) En álgebra abstracta, el álgebra libre es el análogo no conmutativo del anillo de polinomios. Sea R un anillo. El álgebra libre en n indeterminadas, X1..., Xn, es el anillo generado por todas las combinaciones lineales de los productos de las variables. Este anillo es denotado por R<X1..., Xn>. A diferencia de un anillo polinómico, las variables no conmutan. Por ejemplo X1X2 no es igual a X2X1. * Datos: Q5500180 (es) |
| rdfs:label | Álgebra libre (es) Free algebra (en) Polynôme non commutatif (fr) 自由代数 (ja) |
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