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In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions. The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going down refers to the case when a chain can be extended by "downward inclusion". The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Seidenberg. These are known as the going-up and going-down theorems. == Going up and going down == Let ''A''⊆''B'' be an extension of commutative rings. The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in ''B'', each member of which lies over members of a longer chain of prime ideals in ''A'', to be able to be extended to the length of the chain of prime ideals in ''A''. 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「Going up and going down」の詳細全文を読む スポンサード リンク
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