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In mathematics, a finitely generated algebra is an associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,…,''a''''n'' of ''A'' such that every element of ''A'' can be expressed as a polynomial in ''a''1,…,''a''''n'', with coefficients in ''K''. If it is necessary to emphasize the field ''K'' then the algebra is said to be finitely generated over ''K'' . Algebras that are not finitely generated are called infinitely generated. Finitely generated reduced commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this reason, these algebras are also referred to as (commutative) affine algebras. == Examples == * The polynomial algebra ''K''() is finitely generated. The polynomial algebra in infinitely countably many generators is infinitely generated. * The field ''E'' = ''K''(''t'') of rational functions in one variable over an infinite field ''K'' is ''not'' a finitely generated algebra over ''K''. On the other hand, ''E'' is generated over ''K'' by a single element, ''t'', ''as a field''. * If ''E''/''F'' is a finite field extension then it follows from the definitions that ''E'' is a finitely generated algebra over ''F''. * Conversely, if ''E'' /''F'' is a field extension and ''E'' is a finitely generated algebra over ''F'' then the field extension is finite. This is called Zariski's lemma. See also integral extension. * If ''G'' is a finitely generated group then the group ring ''KG'' is a finitely generated algebra over ''K''. 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「Finitely generated algebra」の詳細全文を読む スポンサード リンク
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