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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 $i_j ∈ I^j$ .

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.

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  • $\begingroup$ Your questions mention neither reduction nor the ideal $J$, is this normal? (no pun intended). $\endgroup$
    – abx
    Commented Mar 1, 2015 at 8:01
  • $\begingroup$ Is this homework? $\endgroup$
    – Dima Pasechnik
    Commented Mar 1, 2015 at 9:25

1 Answer 1

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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).

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