|
|
| Line 1: |
Line 1: |
| {{Otheruses4|free algebras in ring theory|the more general free algebras in [[universal algebra]]|free object}}
| | Howdy. The [http://Imageshack.us/photos/author%27s author's] name is Eusebio remember, though , he never really cherished that name. In his professional life he can be a people manager. He's always loved living located in Guam and he features everything that he prerequisites there. To drive is one of a person's things he loves mainly. He's been working by his website for various time now. Check one out here: http://[http://Answers.yahoo.com/search/search_result?p=prometeu.net&submit-go=Search+Y!+Answers prometeu.net]<br><br>Feel free to surf to my web blog :: [http://prometeu.net Clash Of Clans Hack No Survey No Jailbreak] |
| In [[mathematics]], especially in the area of [[abstract algebra]] known as [[ring theory]], a '''free algebra''' is the noncommutative analogue of a [[polynomial ring]] (which may be regarded as a '''free commutative algebra''').
| |
| | |
| ==Definition==
| |
| For ''R'' a [[commutative ring]], the free ([[associative]], [[unital algebra|unital]]) [[algebra (ring theory)|algebra]] on ''n'' [[indeterminate_(variable)|indeterminate]]s {''X''<sub>1</sub>,...,''X<sub>n</sub>''} is the [[free module|free ''R''-module]] with a basis consisting of all [[Word_(mathematics)|words]] over the alphabet {''X''<sub>1</sub>,...,''X<sub>n</sub>''} (including the empty word, which is the unity of the free algebra). This ''R''-module becomes an [[algebra (ring theory)|''R''-algebra]] by defining a multiplication as follows: the product of two basis elements is the [[concatenation]] of the corresponding words:
| |
| | |
| :<math>\left(X_{i_1}X_{i_2} \cdots X_{i_m}\right) \cdot \left(X_{j_1}X_{j_2} \cdots X_{j_n}\right) = X_{i_1}X_{i_2} \cdots X_{i_m}X_{j_1}X_{j_2} \cdots X_{j_n},</math>
| |
| | |
| and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an ''R''-algebra must be ''R''-bilinear). This ''R''-algebra is denoted ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩. This construction can easily be generalized to an arbitrary set ''X'' of indeterminates.
| |
| | |
| In short, for an arbitrary set <math>X=\{X_i\,;\; i\in I\}</math>, the '''free ([[associative]], [[unital algebra|unital]]) ''R''-[[algebra (ring theory)|algebra]] on ''X''''' is
| |
| :<math>R\langle X\rangle:=\bigoplus_{w\in X^\ast}R w</math>
| |
| with the ''R''-bilinear multiplication that is concatenation on words, where ''X''* denotes the [[free monoid]] on ''X'' (i.e. words on the letters ''X''<sub>i</sub>), <math>\oplus</math> denotes the external [[Direct sum of modules|direct sum]], and ''Rw'' denotes the [[free module|free ''R''-module]] on 1 element, the word ''w''.
| |
| | |
| For example, in ''R''⟨''X''<sub>1</sub>,''X''<sub>2</sub>,''X''<sub>3</sub>,''X''<sub>4</sub>⟩, for scalars ''α,β,γ,δ'' ∈''R'', a concrete example of a product of two elements is <math>(\alpha X_1X_2^2 + \beta X_2X_3)\cdot(\gamma X_2X_1 + \delta X_1^4X_4) = \alpha\gamma X_1X_2^3X_1 + \alpha\delta X_1X_2^2X_1^4X_4 + \beta\gamma X_2X_3X_2X_1 + \beta\delta X_2X_3X_1^4X_4</math>.
| |
| | |
| The non-commutative polynomial ring may be identified with the [[monoid ring]] over ''R'' of the [[free monoid]] of all finite words in the ''X''<sub>''i''</sub>.
| |
| | |
| ==Contrast with Polynomials==
| |
| Since the words over the alphabet {''X''<sub>1</sub>, ...,''X<sub>n</sub>''} form a basis of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩, it is clear that any element of ''R''⟨''X''<sub>1</sub>, ...,''X<sub>n</sub>''⟩ can be uniquely written in the form:
| |
| | |
| :<math>\sum\limits_{i_1,i_2, \cdots ,i_k\in\left\lbrace 1,2, \cdots ,n\right\rbrace} a_{i_1,i_2, \cdots ,i_k} X_{i_1} X_{i_2} \cdots X_{i_k},</math> | |
| | |
| where <math>a_{i_1,i_2,...,i_k}</math> are elements of ''R'' and all but finitely many of these elements are zero. This explains why the elements of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") ''X''<sub>1</sub>,...,''X<sub>n</sub>''; the elements <math> a_{i_1,i_2,...,i_k}</math> are said to be "coefficients" of these polynomials, and the ''R''-algebra ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩ is called the "non-commutative polynomial algebra over ''R'' in ''n'' indeterminates". Note that unlike in an actual polynomial ring, the variables do not [[commutative operation|commute]]. For example ''X''<sub>1</sub>''X''<sub>2</sub> does not equal ''X''<sub>2</sub>''X''<sub>1</sub>.
| |
| | |
| More generally, one can construct the free algebra ''R''⟨''E''⟩ on any set ''E'' of [[generating set|generators]]. Since rings may be regarded as '''Z'''-algebras, a '''free ring''' on ''E'' can be defined as the free algebra '''Z'''⟨''E''⟩.
| |
| | |
| Over a [[field (mathematics)|field]], the free algebra on ''n'' indeterminates can be constructed as the [[tensor algebra]] on an ''n''-dimensional [[vector space]]. For a more general coefficient ring, the same construction works if we take the [[free module]] on ''n'' [[generating set|generators]].
| |
| | |
| The construction of the free algebra on ''E'' is [[functor]]ial in nature and satisfies an appropriate [[universal property]]. The free algebra functor is [[left adjoint]] to the [[forgetful functor]] from the category of ''R''-algebras to the [[category of sets]].
| |
| | |
| Free algebras over [[division ring]]s are [[free ideal ring]]s.
| |
| | |
| ==See also==
| |
| | |
| *[[Cofree coalgebra]]
| |
| *[[Tensor algebra]]
| |
| *[[Free object]]
| |
| *[[Noncommutative ring]]
| |
| | |
| ==References==
| |
| * {{cite book | last1=Berstel | first1=Jean | last2=Reutenauer | first2=Christophe | title=Noncommutative rational series with applications | series=Encyclopedia of Mathematics and Its Applications | volume=137 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-19022-0 | zbl=1250.68007 }}
| |
| * {{springer|id=f/f041520|author=L.A. Bokut'|title=Free associative algebra}}
| |
| | |
| [[Category:Algebras]]
| |
| [[Category:Ring theory]]
| |
| [[Category:Free algebraic structures]]
| |
Howdy. The author's name is Eusebio remember, though , he never really cherished that name. In his professional life he can be a people manager. He's always loved living located in Guam and he features everything that he prerequisites there. To drive is one of a person's things he loves mainly. He's been working by his website for various time now. Check one out here: http://prometeu.net
Feel free to surf to my web blog :: Clash Of Clans Hack No Survey No Jailbreak