Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in degree zero, and nilpotent torsion everywhere else. It's a horrible, non-Noetherian ring.
Question: What is the Goldie dimension (aka uniform dimension) of the ring $\pi_\ast^{(p)}$?
Here, the Goldie dimension is the maximal number $n$ such that there exist $n$ ideals $I_1,\dots,I_n \subseteq \pi_\ast^{(p)}$ such that the induced map of modules $I_1 \oplus \cdots \oplus I_n \to \pi_\ast^{(p)}$ is injective.
Notes:
To say that the Goldie dimension is 1 is to say that any two nonzero ideals have nonzero intersection, so in order to rule out this case it would suffice to find two nonzero ideals in $\pi_\ast^{(p)}$ whose intersection is zero.
For more on Goldie dimension, see Lam's Lectures on Modules and Rings, Chapter 6.
I expect that the answer is probably $\infty$, but maybe not. The answer is $\infty$ if we do not localize at a prime, because $\oplus_p \pi_{\geq 1}^{(p)} \to \pi_\ast^s$ is an injection.
