The chiral ring of the Coulomb branch of a 4D $\mathcal N=2$ 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 $\mathcal N=2$ 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 $\mathcal N=2$ supersymmetric algebras?
I was asked to clarify the definition of the Coulomb branch in non-Lagrangian theories; let's define them for $\mathcal N=2$ SCFT by the fact that $SU(2)_R$ symmetry acts on the Coulomb branch operators trivially.