If two torsion theories on a ring localize the ring to the same extension ring, I can find no reason that their "meet" in the lattice of torsion theories must also localize to the same ring. I cannot find anything in Golan's encyclopedia that addresses questions like this.
Does anyone have a counter-example?
Here is a weaker question, not directly related to torsion theories.
Is there an example of the following:
A ring homomorphism R $\to$ S , S-modules P and Q , R-monomorphisms M $\to$ P and M $\to$ Q such that the image of each is an essential R-submodule, but such that the image of M in P $\times$ Q has no essential extension within the product that is an S-submodule