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 Groebner basis theory is not yet powerful enough to deal with non-associative multiplication. So far in literature there aren't any general theories to solve this problem. This paper investigates how to extend the Groebner basis theory to indexed polynomials. Firstly the indexed polynomials are described via an infinitely generated free commutative monoid ring. We then provide a Groebner basis of the defining syzygy set in each restricted ring, and prove that the canonical form of a Riemann tensor indexed polynomial is independent of restricted rings. Finally we find out the minimal restricted ring. Due to the restricted ring, the canonical form of an indexed polynomial can be obtained from a reduction with respect to the Groebner basis.