# examples of radicals of ideals in commutative rings

Let $R$ be a commutative ring. Recall, that ideals $I,J$ in $R$ are called iff $I+J=R$. It can be shown, that if $I,J$ are coprime, then $IJ=I\cap J$. Elements $x_{1},\ldots,x_{n}\in R$ are called pairwise coprime iff $(x_{i})+(x_{j})=R$ for $i\neq j$. It follows by induction, that for pairwise coprime $x_{1},\ldots,x_{n}\in R$ we have $(x_{1}\cdots x_{n})=(x_{1})\cap\cdots\cap(x_{n})$,

Let $x\in R$ be such that

 $x=p_{1}^{\alpha_{1}}\cdots p_{n}^{\alpha_{n}},$

for some prime elements $p_{i}\in R$, $\alpha_{i}\in\mathbb{N}$ and assume that $p_{1},\ldots,p_{n}$ are coprime. Denote by

 $\overline{x}=p_{1}\cdots p_{n}.$

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

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

Proof. ,,$\supseteq$” Let $\alpha=\mathrm{max}(\alpha_{1},\ldots,\alpha_{n})$. Then we have

 $\overline{x}^{\alpha}=(p_{1}\cdots p_{n})^{\alpha}=p_{1}^{\alpha}\cdots p_{n}^% {\alpha}=p_{1}^{\alpha-\alpha_{1}}\cdots p_{n}^{\alpha-\alpha_{n}}p_{1}^{% \alpha_{1}}\cdots p_{n}^{\alpha_{n}}=yx$

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

,,$\subseteq$” Assume that $y\in r\big{(}(x)\big{)}$ and $y\neq 0$. Then there is $n\in\mathbb{N}$ such that $y^{n}\in(x)$. Thus $x$ divides $y^{n}$. Of course for any $i\in\{1,\ldots,n\}$ 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 $i\neq j$ elements $p_{i}$ and $p_{j}$ are coprime, thus $\overline{x}$ divides $y$ and therefore $y\in(\overline{x})$, which completes the proof. $\square$

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 $p_{1}\cdots p_{n}$, where for $i\neq j$ elements $p_{i}$ and $p_{j}$ are not associated primes.

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

 $r\big{(}(12)\big{)}=(6);$
 $r\big{(}(9)\big{)}=(3);$
 $r\big{(}(7)\big{)}=(7);$
 $r\big{(}(1125)\big{)}=(15).$
Title examples of radicals of ideals in commutative rings ExamplesOfRadicalsOfIdealsInCommutativeRings 2013-03-22 19:04:34 2013-03-22 19:04:34 joking (16130) joking (16130) 5 joking (16130) Example msc 16N40 msc 14A05 msc 13-00