The chiral ring of the Coulomb branch of a 4d N=2 supersymmetric gauge theory is given by the Casimirs of the vector multiplet scalars, and they don't have non-trivial relations; the Casimirs are always independent.
Also in Gaiotto's class of N=2 non-Lagrangian theories, the chiral ring of the Coulomb branch doesn't (seem to) have relations.
Is it a general fact? If so, how can we deduce it from the N=2 susy algebras?
I was asked to clarify the definition of the Coulomb branch in non-Lagrangian theories; let's define them for N=2 SCFT by the fact that $SU(2)_R$ symmetry acts on the Coulomb branch operators trivially.