# If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties:

• $I_n$ is generated by a (finite) number of polynomials which are homogeneous of degree 2 and which are no monomials.
• If a variable $x_{i}$ occurs in one of the generating polynomials, it occurs only once and with degree one.
• Two distinct generating polynomials do not share the same monomial.

Question: is $I_{n}$ radical?

What I have so far: the above properties show that a generating polynomial is squarefree and in fact irreducible, hence it generates a prime ideal as we are working in a UFD. As the sum of two radical ideals is not radical in general, I do not know of a way to continue.

Let $R=\mathbb C[x_1,\dots,x_9]$ and $I=(x_1x_9-x_4x_8,x_4x_6-x_7x_9,x_2x_5-x_3x_9,x_2x_3-x_5x_6).$
Macaulay2 gives $$\sqrt I=(x_4x_8-x_1x_9, x_4x_6-x_7x_9,x_2x_5-x_3x_9,x_2x_3-x_5x_6, x_1x_5x_6^2-x_5x_6x_7x_8).$$