|
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a Noetherian ring. The theorem is sometimes referred to by its German name, ''Krulls Hauptidealsatz'' (''Satz'' meaning "proposition" or "theorem"). Formally, if ''R'' is a Noetherian ring and ''I'' is a principal, proper ideal of ''R'', then ''I'' has height at most one. This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem. This says that if ''R'' is a Noetherian ring and ''I'' is a proper ideal generated by ''n'' elements of ''R'', then ''I'' has height at most ''n''. The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theorem of dimension theory. Bourbaki's ''Commutative Algebra'' gives a direct proof. Kaplansky's ''Commutative ring'' includes a proof due to David Rees. ==References== * , see in particular section (12.I), p. 77 *http://www.math.lsa.umich.edu/~hochster/615W10/supDim.pdf 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「Krull's principal ideal theorem」の詳細全文を読む スポンサード リンク
|