Polynomial ergodic averages for certain countable ring actions

A recent result of Frantzikinakis establishes sufficient conditions for joint ergodicity in the setting of $\mathbb{Z}$-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action $(T_n)_{n \in F}$ of a countable field $F$ with characteristic zero on a probability space $(X,\mathcal{B},\mu)$ and a family $\{p_1,\dots,p_k\}$ of independent polynomials, we have \[ \lim_{N \to \infty} \frac{1}{|\Phi_N|}\sum_{n \in \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod_{j=1}^k \int_X f_i \ d\mu,\] where $f_i \in L^{\infty}(\mu)$, $(\Phi_N)$ is a F\o lner sequence of $(F,+)$, and the convergence takes place in $L^2(\mu)$. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.