  We consider $\mathcal N=2$ $SU(2)$ gauge theories in four dimensions (pure or
mass deformed) and discuss the properties of the simplest chiral observables in
the presence of a generic $\Omega$-deformation. We compute them by equivariant
localization and analyze the structure of the exact instanton corrections to
the classical chiral ring relations. We predict exact relations valid at all
instanton number among the traces $\langle\text{Tr}\varphi^{n}\rangle$, where
$\varphi$ is the scalar field in the gauge multiplet. In the
Nekrasov-Shatashvili limit, such relations may be explained in terms of the
available quantized Seiberg-Witten curves. Instead, the full two-parameter
deformation enjoys novel features and the ring relations require non trivial
additional derivative terms with respect to the modular parameter. Higher rank
groups are briefly discussed emphasizing non-factorization of correlators due
to the $\Omega$-deformation. Finally, the structure of the deformed ring
relations in the $\mathcal N=2^{\star}$ theory is analyzed from the point of
view of the Alday-Gaiotto-Tachikawa correspondence proving consistency as well
as some interesting universality properties.
