  Given a finite group $\Gamma$, a regular $t$-balanced Cayley map (RBCM$_{t}$
for short) is a regular Cayley map $\mathcal{CM}(G,\Omega,\rho)$ such that
$\rho(\omega)^{-1}=\rho^{t}(\omega)$ for all $\omega\in\Omega$.
  In this paper, we clarify a connection between quotients of polynomial rings
and RBCM$_{t}$'s on abelian groups, so as to propose a new approach for
classifying RBCM$_{t}$'s. We obtain many new results, in particular, a complete
classification for RBCM$_{t}$'s on abelian 2-groups.
