Definition (Reduction Ideal). Let $ I $ and $ J $ be ideals of $ R $. Then $ J $ is called a reduction of $ I $ iff $ J \subseteq I $ and there exists an $ n \in \mathbb{N} $ such that $ I^{n} = J I^{n - 1} $.

Let $ R $ be a ring and $ I,J $ ideals of $ R $ with $ J \subseteq I $ and $ I $ finitely generated.

Questions:

1. $ J $ is a reduction of $ I $ iff $ \text{rad}(J) $ is a reduction of $ \text{rad}(I) $. (Note that if $ J $ is a reduction of $ I $, then $ \text{rad}(J) = \text{rad}(I) $.)

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

3. The integral closure of a homogeneous ideal is homogeneous.