It is one of the oldest research topics in computer algebra to determine the equivalence of Riemann tensor indexed polynomials. However, it remains to be a challenging problem since Gr¨obner basis theory is not yet powerful enough to deal with ideals that cannot be finitely generated. This paper solves the problem by extending Gr¨obner basis theory. First, the polynomials are described via an infinitely generated free commutative monoid ring. The authors then provide a decomposed form of the Gr¨obner basis of the defining syzygy set in each restricted ring. The canonical form proves to be the normal form with respect to the Gr¨obner basis in the fundamental restricted ring, which allows one to determine the equivalence of polynomials. Finally, in order to simplify the computation of canonical form, the authors find the minimal restricted ring.