|
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 on ''n'' indeterminates is the free ''R''-module with a basis consisting of all words over the alphabet (including the empty word, which is the unity of the free algebra). This ''R''-module becomes an ''R''-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words: : 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''1,...,''Xn''⟩. This construction can easily be generalized to an arbitrary set ''X'' of indeterminates. In short, for an arbitrary set , the free (associative, unital) ''R''-algebra on ''X'' is : 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''i), denotes the external direct sum, and ''Rw'' denotes the free ''R''-module on 1 element, the word ''w''. For example, in ''R''⟨''X''1,''X''2,''X''3,''X''4⟩, for scalars ''α,β,γ,δ'' ∈''R'', a concrete example of a product of two elements is . 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''''i''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Free algebra」の詳細全文を読む スポンサード リンク
|