Computing Intersection of Ideals

Question

I am having hard time computing the intersection of ideals. If there is a graph G=(V(G),E(G))such that

(vertex set) v(G)={$x_{1}$,$x_{2}$,$x_{3}$,$x_{4}$,$x_{5}$},

(edge set) E(G)={$x_{1}x_{2},x_{2}x_{3},x_{3}x_{4},x_{4}x_{5},x_{1}x_{5}x_{2}x_{5},x_{1}x_{4}$},

then we can express the edge ideal J(G) as follows:

J(G)= <$x_{1}x_{2}$> $\cap$ <$x_{2}x_{3}$> $\cap$ <$x_{3}x_{4}$> $\cap$ <$x_{4}x_{5}$> $\cap$ <$x_{1}x_{5}$> $\cap$ <$x_{2}x_{5}$> $\cap$ <$x_{1}x_{4}$>

If we calculate this, we some how get

J(G)= <$x_{1}x_{3}x_{5},x_{1}x_{2}x_{4},x_{2}x_{4}x_{5}>$

I was wondering how I can compute those intersection to get the final result.

I know that we take the lcm of component ideal of monomial ideals, but I have no idea how to apply it.

Thank you very much!

Answer 1

$\langle x_1x_2 \rangle \cap \langle x_2x_3 \rangle = \langle x_1x_2x_3 \rangle $

$\langle x_1x_2x_3 \rangle \cap \langle x_3x_4 \rangle = \langle x_1x_2x_3x_4 \rangle $

$\langle x_1x_2x_3x_4\rangle \cap \langle x_4x_5 \rangle = \langle x_1x_2x_3x_4x_5 \rangle $

Then the remaining terms don't change anything, so $\langle x_1x_2x_3x_4x_5 \rangle $ is the correct ideal.

In particular, your first and third proposed generators aren't in $\langle x_1x_2 \rangle $ and your second proposed generator isn't in $\langle x_2x_3 \rangle $.