# Properties of Integral Closure [closed]

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.

## closed as off-topic by Dima Pasechnik, Alex Degtyarev, Peter Crooks, Karl Schwede, Joonas IlmavirtaMar 1 '15 at 19:42

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• Your questions mention neither reduction nor the ideal $J$, is this normal? (no pun intended). – abx Mar 1 '15 at 8:01
• Is this homework? – Dima Pasechnik Mar 1 '15 at 9:25

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