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Let $R$ be a ring, commutative with $1$, let $\mathfrak{i}$ be an ideal, not the whole ring. In general $\mathfrak{i}^2\subseteq\mathfrak{i}$. Can this inclusion be an equality, or it is always a strict inclusion?

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3 Answers 3

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1) A rich source of examples for ideals $I\subset R$ satisfying $I^2=I$ is to take for $R$ an arbitrary product $$R=\prod_{t\in T} R_t$$ of arbitrary rings $R_t$, to choose an arbitrary subset $S\subset T$ and to take for $I$ the ideal $$I=\prod_{t\in T} I_i$$ where $I_t=0$ if $t\in S$ and $I_i=R_i$ if $t\in T\setminus S$.

2) For finitely generated ideals $I\subset R$ (an empty restriction for noetherian rings) we have a complete classification of these ideals :

Theorem
If a finitely generated ideal $I\subset R$ satisfies $I=I^2$, then $I$ is principal, generated by an idempotent : $$I=Re,\quad e=e^2\in R$$
Proof
Since $I=I\cdot I$, Nakayama tells us that there exists $e\in I$ such that for all $i\in I$ we have $i=ei$.
This immmediately implies (by putting $i=e$) and that $e^2=e$ and $I=Re$.

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  • $\begingroup$ What if $I$ is a prime ideal? $\endgroup$ Apr 2, 2013 at 7:27
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    $\begingroup$ There are many examples with $I$ prime: you can take in my class 1) of examples: $S=\{t_0\}=$ a singleton subset of $T$ and $R_{t_0}=$ some field. The corresponding $I$ then will be prime, actually even maximal. $\endgroup$ Apr 2, 2013 at 7:37
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    $\begingroup$ So do i need to require $R$ Dedekind, to say that $\mathfrak{i}\neq\mathfrak{i}^2$ for every non-zero prime ideal $\mathfrak{i}$? $\endgroup$ Apr 2, 2013 at 7:47
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    $\begingroup$ No, Dedekind is too strong. In view of the theorem in my answer it is sufficient to require that $R$ be a noetherian domain. Indeed, a domain has $0$ and $1$ as only idempotents, so that $I=I^2$ implies $I=(0)$ (which you excluded) or $I=(1)=R$ (which is not prime). $\endgroup$ Apr 2, 2013 at 7:54
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Another uninteresting, trivial case to rule out is $\mathfrak{i}=(0)$, of course.

Here is a more interesting example: let $R=\mathbb{Z}[\sqrt{2},\sqrt[3]{2},\sqrt[4]{2},\ldots]$ and $\mathfrak{i}=(2,\sqrt{2},\sqrt[3]{2},\ldots)$. Then any element of $\mathfrak{i}$ is a finite $\mathbb{Z}$-linear combination of roots of $2$, which is of course also a finite $\mathbb{Z}$-linear combination of products of (higher) roots of $2$, e.g. $$5\cdot\sqrt{2}-\sqrt[3]{2}=5\cdot(\sqrt[4]{2}\cdot\sqrt[4]{2})-(\sqrt[6]{2}\cdot\sqrt[6]{2}),$$ so $\mathfrak{i}\subseteq\mathfrak{i}^2$, and hence $\mathfrak{i}=\mathfrak{i}^2$. No element of $\mathfrak{i}$ is a unit so we know that $\mathfrak{i}\neq R$.

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    $\begingroup$ This example has some very interesting features: the ideal is not principal, it is maxiamal and above all the ring is a domain, which is surprising. $\endgroup$ Apr 2, 2013 at 7:28
  • $\begingroup$ @GeorgesElencwajg This is also an example of an idempotent ideal which is not finitely generated. $\endgroup$
    – user26857
    Apr 2, 2013 at 14:22
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If $\mathfrak{i}$ is generated by an idempotent then $\mathfrak{i}^2=\mathfrak{i}$.

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