Composed Groebner basis over Noetherian domain

For Groebner basis in n variables and composition in m(m≥n) variables in a polynomial ring over Noetherian domain,it is proved that Groebner basis computation and composition is commutative if composition is compatible with two term orderings on the different polynomial rings and composition is a lists of monic polynomials with its leading powering products is the products of permuted powering and powering products of other remained variables by using S-polynomial and syzygy condition.Therefore,minimal Groebner basis computation is also commutative with composition under this condition.Especially,Groebner basis computation and composition is commutative if composition is compatible with term orderings and composition is a lists of monic polynomials with its leading powering product is a permuted powering when m=n.