Радикал идеала

В коммутативной алгебре радикал идеала I — это идеал, образованный всеми элементами x такими, что некоторая степень x принадлежит I. Радикальный идеал — это идеал, совпадающий со своим собственным радикалом.

Определение

Радикал идеала I в коммутативном кольце R, обозначаемый ${\sqrt {I}}$ , определяется как

{\sqrt {I}}=\{r\in R\mid \exists n\in \mathbb {N} \,\,\,r^{n}\in I\}

Интуитивно, для получения радикала идеала нужно взять корни всех возможных степеней из его элементов. Эквивалентное определение радикала идеала I — это прообраз нильрадикала $R/I$ при отображении факторизации. Это также доказывает, что ${\sqrt {I}}$ является идеалом.

Примеры

В кольце целых чисел радикал главного идеала $(a)$ — это идеал, порождённый произведением всех простых делителей $a$ .
Радикал примарного идеала прост. Если радикал идеала максимален, то этот идеал примарен (если же радикал прост, то идеал не обязательно примарен).
В любом коммутативном кольце ${\sqrt {P^{n}}}=P$ для простого идеала $P$ ^[1]. В частности, каждый простой идеал радикален.

Свойства

${\sqrt {\sqrt {I}}}={\sqrt {I}}$ . Более того, ${\sqrt {I}}$ — это наименьший радикальный идеал, содержащий I.
${\sqrt {I}}$ — это пересечение всех простых идеалов, содержащих I. В частности, нильрадикал — это пересечение всех простых идеалов.
Идеал является радикальным тогда и только тогда, когда факторкольцо по нему не содержит нетривиальных нильпотентов.

Приложения

Основная мотивация для изучения радикалов — это их появление в знаменитой теореме Гильберта о нулях из коммутативной алгебры. Наиболее простая формулировка этой теоремы имеет следующий вид: для любого алгебраически замкнутого поля $k$ и любого конечнопорождённого идеала $J$ в кольце многочленов от $n$ переменных над полем $k$ верно следующее равенство:

\operatorname {I} (\operatorname {V} (J))={\sqrt {J}},

где

\operatorname {V} (J)=\{x\in k^{n}\ |\ f(x)=0~\forall f\in J\}

\operatorname {I} (S)=\{f\in k[x_{1},x_{2},\ldots x_{n}]\ |\ f(x)=0~\forall x\in S\}.

Примечания

↑ Атья и Макдональд, 2003, Предложение 4.2.

Литература

Атья М., Макдональд И. . Введение в коммутативную алгебру. — М.: Факториал Пресс, 2003. — ISBN 5-88688-067-4.
Eisenbud, David. . Commutative Algebra with a View Toward Algebraic Geometry. — Springer-Verlag, 1995. — (Graduate Texts in Mathematics, vol. 150). — ISBN 0-387-94268-8.

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