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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 ringsand RBCM$_{t}$'s on abelian groups, so as to propose a new approach forclassifying RBCM$_{t}$'s. We obtain many new results, in particular, a completeclassification for RBCM$_{t}$'s on abelian 2-groups.

Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups