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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.

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Apologies for this possibly diversionary comment - can you recommend some reference starting point (hopefully pedagogical!) for this topic for beginning students? I have seen some papers and had worked on a project involving the calculation of the chiral ring but could never get my hands on a very clear exposition of the topic - it was quite hazy to pick it up from cutting edge papers! –  user6818 Dec 10 '11 at 20:21
Which topic are you referring to? Chiral rings? This paper arxiv.org/abs/hep-th/0211170 contains a nice review. In general, read and understand all papers by Seiberg or by Witten. Problem solved. –  Yuji Dec 11 '11 at 6:52
There's a reason why textbooks and papers are called differently:) I say, you should just try reading a paper which interests you most, using various references. If you can, then you're ready; if you can't, then you're not. –  Yuji Dec 15 '11 at 10:46
Well, anything which was done in the last century should be considered basic. The SUSY textbook by Terning covers basic stuffs pretty well. –  Yuji Dec 16 '11 at 3:05
Thanks for the comments. I am not sure how to go about learning everything "done in the last century" - that sounds scary! But may be I can try reading Terning - though that is a very terse book! –  user6818 Dec 20 '11 at 21:41
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