高度 (環論)

在交換代數中，一個環 $R$ 的理想 $I$ 的高度是包含於 $I$ 的素理想鏈長度之上確界。

素理想鏈及其長度的定義如下：設交換環 $R$ 中有 $n+1$ 個素理想 $P_{0},\ldots ,P_{n}$ ，使得

P_{0}\subsetneq P_{1}\subsetneq \ldots \subsetneq P_{n}

則稱之為長度為 $n$ 的素理想鏈。若 $P_{n}\subset I$ ，則稱此鏈包含於 $I$ 。一個無法插入新的素理想的鏈被稱作極大的

在代數幾何中，這可以詮釋為閉子概形 $\mathrm {Spec} (R/I)\subset \mathrm {Spec} (R)$ 的餘維度。

在諾特環的情形，Krull 高度定理斷言：由 $n$ 個元素生成的理想其高度必 $\leq n$ 。

文獻

H. Matsumura, Commutative algebra. ISBN 0-8053-7026-9.

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