In [FGRS1,FGRS2] the relationship between universal and elementary theory of a group ring $R[G]$ corresponding associated $G$ $R$ was examined. Here we assume that is commutative with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings respectively. Axiom systems were provided in [FGRS1]. [FGRS1] it proved if elementarily equivale...