In this study, we generalize the rings of the form $F_2+uF_2+...+u^kF_2$ and $F_2+uF_2+vF_2+uvF_2$ to a family of rings that we denote by $R_{k,m}$, where $R_{k,m}=F_2[u,v]/\langle u^k,v^m,uv-vu\rangle$. We find a duality-preserving Gray map from $R_{k,m}$ to $R_2^{k,m}$, and using some of the common construction methods of self-dual codes we find many good binary self-dual codes as the Gray images of self-dual codes over $R_{k,m}$ for suitable $k$ and $m$.