阿廷環

阿廷環是抽象代數中一類滿足降鏈條件的環，以其開創者埃米爾·阿廷命名。

定義

一個環稱作阿廷環，若且唯若對每個由的理想構成的降鏈，必存在，使得對所有的都有（換言之，此降鏈將會固定）。

將上述定義中的理想代換為左理想或右理想，可以類似地定義左阿廷環與右阿廷環，A是左（右）阿廷環若且唯若A在自己的左（右）乘法下形成一個左（右）阿廷模；對於交換環則無須分別左右。

例子

  • 設為一個域，若環是佈於上的有限維代數，則是阿廷環。

基本性質

若一個環是交換阿廷環，則滿足下列性質：

  • 是諾特環。
  • 每個素理想皆是極大理想。
  • 僅有有限個素理想。
  • 對每個素理想的局部化誘導出同構 。

就代數幾何的觀點，阿廷環的譜在拓樸上只是有限多個點，但其結構層可能帶有冪零的元素，這就使得局部阿廷環成為描述無窮小變化量的代數語言。

参见条目

文獻

  • Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712--730.
  • Serge Lang, Algebra (2002), Graduate Texts in Mathematics 211, Springer. ISBN 0-387-95385-X

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