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In mathematics, a commutative ring ''R'' is catenary if for any pair of prime ideals :''p'', ''q'', any two strictly increasing chains :''p''=''p''0 ⊂''p''1 ... ⊂''p''''n''= ''q'' of prime ideals are contained in maximal strictly increasing chains from ''p'' to ''q'' of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain ''n'' is usually the difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings. The word 'catenary' is derived from the Latin word ''catena'', which means "chain". There is the following chain of inclusions. ==Dimension formula== Suppose that ''A'' is a Noetherian domain and ''B'' is a domain containing ''A'' that is finitely generated over ''A''. If ''P'' is a prime ideal of ''B'' and ''p'' its intersection with ''A'', then : The dimension formula for universally catenary rings says that equality holds if ''A'' is universally catenary. Here κ(''P'') is the residue field of ''P'' and tr.deg. means the transcendence degree (of quotient fields). In fact, when ''A'' is not universally catenary, but , then equality also holds. 〔http://www.math.lsa.umich.edu/~hochster/615W14/615.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「catenary ring」の詳細全文を読む スポンサード リンク
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