Given a set of monomials $\{G_1,\ldots,G_t\}$ generating a non-null monomial ideal $I \le K[X_1,\ldots,K_n]$ I would like to check that they form a Gröbner basis.
This is done by Buchberger's algorithm by computing semizygies $S(X^\alpha,X^\beta) = X^{\max{\alpha,\beta}-\alpha} X^\alpha - X^{\max{\alpha,\beta}-\beta}X^\beta = 0$. The problem is that then I should compute $R(0;G)$ the remainder of zero by polynomial division by the elements of $G$. However the polynomial division that I know is not defined for zero.
How can I solve this problem?