On groups with cubic polynomial conditions

Let FdFd be the free group of rank d  , freely generated by {y1,…,yd}{y1,…,yd}, and let DFdDFd be the group ring over an integral domain DD. Given a subset EdEd of FdFd containing the generating set, assign to each s   in EdEd a monic polynomial ps(x)=xn+cs,n−1xn−1+…+cs,1x+cs,0∈D[x]ps(x)=xn+cs,n−1xn−1+…+cs,1x+cs,0∈D[x] and define the quotient ringA(d,n,Ed)=DFd〈ps(s)|s∈Ed〉ideal. When ps(s)ps(s) is cubic for all s  , we construct a finite set EdEd such that A(d,n,Ed)A(d,n,Ed) has finite rank over an extension of DD by inverses of some of the coefficients of the polynomials. When the polynomials are all equal to (x−1)3(x−1)3 and D=Z[16], we construct a finite subset PdPd of FdFd such that the quotient ring A(d,3,Pd)A(d,3,Pd) has finite DD-rank and its augmentation ideal is nilpotent. The set P2P2 is {y1,y2,y1y2,y1−1y2,y12y2,y1y22,[y1,y2]} and we prove that (x−1)3=0(x−1)3=0 is satisfied by all elements in the image of F2F2 in A(2,3,Pd)A(2,3,Pd).