| 翻訳と辞書 |
nil ideal
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.〔, p. 194〕〔, Definition (b), p. 13〕
The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture.
==Commutative rings==
In a commutative ring, the set of all nilpotent elements forms an ideal known as the nilradical of the ring. Therefore, an ideal of a commutative ring is nil if, and only if, it is a subset of the nilradical; that is, the nilradical is the ideal maximal with respect to the property that each of its elements is nilpotent.
In commutative rings, the nil ideals are more well-understood compared to the case of noncommutative rings. This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent. For instance, if ''a'' is a nilpotent element of a commutative ring ''R'', ''a''·''R'' is an ideal that is in fact nil. This is because any element of the principal ideal generated by ''a'' is of the form ''a''·''r'' for ''r'' in ''R'', and if ''a''n = 0, (''a''·''r'')n = ''a''n·''r''n = 0. It is not in general true however, that ''a''·''R'' is a nil (one-sided) ideal in a noncommutative ring, even if ''a'' is nilpotent.
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「nil ideal」の詳細全文を読む
スポンサード リンク
| 翻訳と辞書 : 翻訳のためのインターネットリソース |
Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.
|
|