Decomposing an ideal using Macaulay2

Question

I give Macaulay2 the ideal $I=(y^2, x) \in Q[x , y]$ and then I put decompose I. The result is $(x , y)$ but I do not understand why. Does it mean that $I = (x , y)$? but that is not true, because we can not create $y$ in $I$.

Answer 1

From the documentation:

decompose is a synonym for minimalPrimes.

This function computes the minimal associated primes of the ideal I using characteristic sets. Geometrically, it decomposes the algebraic set defined by I.

So $(x,y)$ is the minimal associated prime of $I = (y^2,x)$. (It's just the radical of $I$, since $I$ is a primary ideal.)