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Homeradical of an ideal

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# radical of an ideal

Let $R$ be a commutative ring. For any ideal $I$ of $R$, the radical of $I$, written $\sqrt{I}$ or $\operatorname{Rad}(I)$, is the set

$\{a\in R\mid a^{n}\in I\text{ for some integer }n>0\}$ |

The radical of an ideal $I$ is always an ideal of $R$.

If $I=\sqrt{I}$, then $I$ is called a radical ideal.

Every prime ideal is a radical ideal. If $I$ is a radical ideal, the quotient ring $R/I$ is a ring with no nonzero nilpotent elements.

More generally, the radical of an ideal in can be defined over an arbitrary ring. Let $I$ be an ideal of a ring $R$, the radical of $I$ is the set of $a\in R$ such that every m-system containing $a$ has a non-empty intersection with $I$:

$\sqrt{I}:=\{a\in R\mid\mbox{if }S\mbox{ is an $m$-system, and }a\in S,\mbox{ % then }S\cap I\neq\varnothing\}.$ |

Under this definition, we see that $\sqrt{I}$ is again an ideal (two-sided) and it is a subset of $\{a\in R\mid a^{n}\in I\mbox{ for some integer }n>0\}$. Furthermore, if $R$ is commutative, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “generalization” of the radical of an ideal in a commutative ring.

## Mathematics Subject Classification

14A05*no label found*16N40

*no label found*13-00

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