 Powers of an ideal

ID: powers-of-an-ideal

For an ideal

I

R

, put

I^{0} = R

and let

I^{n}

for

n \geq 1

n

factors equal to

I

. It consists of finite sums of products of

n

elements of

I

. These ideals form a descending chain

R \supseteq I \supseteq I^{2} \supseteq \dots

Every positive power has the same radical of an ideal as

I

I^{n} = I (n \geq 1) .

One inclusion follows from

I^{n} \subseteq I

. Conversely, if

x^{m} \in I

, then

x^{mn} \in I^{n}

, giving the other inclusion. Different ideals can therefore define the same radical ideal, even when their successive powers are all distinct.

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