On relationships between algebraic properties of groups and rings in some model-theoretic contexts

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $\omega$-categorical rings as well as about small, $nm$-stable compact $G$-rings, and we also obtain surprising relationships between some conjectures concerning small profinite groups.