examples of radicals of ideals in commutative rings

Let $R$ be a commutative ring. Recall, that ideals $I,J$ in $R$ are called coprime iff $I+J=R$. It can be shown, that if $I,J$ are coprime, then $IJ=I\cap J$. Elements $x_1,\mathrm{\dots },x_n\in R$ are called pairwise coprime iff $(x_i)+(x_j)=R$ for $i\ne j$. It follows by induction, that for pairwise coprime $x_1,\mathrm{\dots },x_n\in R$ we have $(x_1\mathrm{\cdots }{x}_n)=(x_1)\cap \mathrm{\cdots }\cap (x_n)$,

Let $x\in R$ be such that

$$x=p_1^{{\alpha }_1}\mathrm{\cdots }{p}_n^{{\alpha }_n},$$

for some prime elements $p_i\in R$, ${\alpha }_i\in \mathbb{N}$ and assume that $p_1,\mathrm{\dots },p_n$ are coprime. Denote by

$$\overline{x}=p_1\mathrm{\cdots }{p}_n.$$

We shall denote by $r(I)$ the radical of an ideal $I\subseteq R$.

Proposition. $r\left((x)\right)=(\overline{x})$.

Proof. ,,$\supseteq$” Let $\alpha =\max ({\alpha }_1,\mathrm{\dots },{\alpha }_n)$. Then we have

$${\overline{x}}^{\alpha }={(p_1\mathrm{\cdots }{p}_n)}^{\alpha }=p_1^{\alpha }\mathrm{\cdots }{p}_n^{\alpha }=p_1^{\alpha -{\alpha }_1}\mathrm{\cdots }{p}_n^{\alpha -{\alpha }_n}{p}_1^{{\alpha }_1}\mathrm{\cdots }{p}_n^{{\alpha }_n}=yx$$

and thus ${\overline{x}}^{\alpha }\in (x)$. This shows the first inclusion.

,,$\subseteq$” Assume that $y\in r\left((x)\right)$ and $y\ne 0$. Then there is $n\in \mathbb{N}$ such that $y^n\in (x)$. Thus $x$ divides $y^n$. Of course for any $i\in \{1,\mathrm{\dots },n\}$ we have that $p_i$ divides $x$. Thus $p_i$ divides $y^n$ and since $p_i$ is prime, we obtain that $p_i$ divides $y$. Now for $i\ne j$ elements $p_i$ and $p_j$ are coprime, thus $\overline{x}$ divides $y$ and therefore $y\in (\overline{x})$, which completes the proof. $\mathrm{\square }$

Remark. If we assume that $R$ is a PID (and thus UFD), then the previous proposition gives us the full characterization of radicals of ideals in $R$. In particular an ideal in PID is radical if and only if it is generated by an element of the form $p_1\mathrm{\cdots }{p}_n$, where for $i\ne j$ elements $p_i$ and $p_j$ are not associated primes.

Examples. Consider ring of integers $\mathbb{Z}$. Then we have:

$$r\left((12)\right)=(6);$$

$$r\left((9)\right)=(3);$$

$$r\left((7)\right)=(7);$$

$$r\left((1125)\right)=(15).$$

Title	examples of radicals of ideals in commutative rings
Canonical name	ExamplesOfRadicalsOfIdealsInCommutativeRings
Date of creation	2013-03-22 19:04:34
Last modified on	2013-03-22 19:04:34
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	5
Author	joking (16130)
Entry type	Example
Classification	msc 16N40
Classification	msc 14A05
Classification	msc 13-00