
> 
> **Definition(Integral closure)**: Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisﬁes a monic equation
>  $x^n + i\_1x^{n−1} + ··· + i\_n = 0$ such that $i\_j ∈ I^j$ .
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> 


Let $ R $ be a ring and $ I $ ideals of $ R $ and $ I $ be a finitely generated.


**Questions:**


1. The integral closure of $ \text{rad}(I) $ is equal to the radical of the integral closure of $ I $.
2. The integral closure of a homogeneous ideal is homogeneous.



Properties of Integral Closure
1) is obvious : a radical ideal is integrally closed, so $I\subset \mathrm{rad}(I)$ gives $\bar{I}\subset \mathrm{rad}(I)$, hence $\mathrm{rad}(\bar{I})=\mathrm{rad}(I)$. 


2) is more subtle. This is Corollary 5.2.3 in "Integral Closure of Ideals, Rings, and Modules" by Swanson and Huneke, London Mathematical Society Lecture Note Series 336. I recommend that you look at that book for further questions about integral closure (it is available on the web). 


