a set of non-zero polynomials which generate a two-sided ideal $I=\langle G\rangle\subseteq K\langle X\rangle$. Then the set $G$ is a ${\sigma}$-Gröbner basis of $I$ if and only if, for each non-trivial obstruction ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in\bigcup_{1\leq i\leq j\leq s}{\rm NTObs}(i,j)$, its S-polynomial $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has a Gröbner representation in terms of $G$.

This follows directly from Proposition \[sec2pro5\] and Lemmas \[sec2lem6\] and \[sec2lem8\]. In view of Lemma \[sec2lem8\], it suffices to consider each obstruction with overlap, which is either a non-trivial obstruction or a multiple of a non-trivial obstruction. Further, Lemma \[sec2lem6\] treats a multiple of a non-trivial obstruction via the corresponding non-trivial obstruction. Therefore, it is sufficient to consider only non-trivial obstructions.

The Buchberger Criterion enables us to formulate the following procedure for computing Gröbner bases of two-sided ideals. Note that, in the procedure, by a *fair strategy* we mean a selection strategy which ensures that every obstruction is selected eventually. Since these Gröbner bases need not be finite, we have to content ourselves with an enumerating procedure.

\[sec2the11\] Let $s\geq 1$, and let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle$ be a set of non-zero polynomials which generate a two-sided ideal $I=\langle G\rangle\subseteq K\langle X\rangle$. Consider the following sequence of instructions.

-   Let $B=\bigcup_{1\leq i\leq 