On Ideals Generated by R-Circuits

Let R be a commutative ring, and let S=R[Y1,…,Yn] denote the polynomial ring in n variables Y1,…,Yn. We introduce an admissible term order on the monomials in these variables and extend the classical definition of circuits to the case where coefficients lie in a commutative ring R. We investigate the conditions under which an ideal of R[Y1,…,Yn], generated by linear forms, can be generated by R-circuits. Specific results are obtained when R=K[x1,…,xm], which is the polynomial ring over a field K.