$\lc_{\sigma}(f)=c_1$ is called the *leading coefficient*. We let $\lm_{\sigma}(f)=c_1w_1$ and call it the *leading monomial* of $f$.

-   Let $I\subseteq K\langle X\rangle$ be a two-sided ideal. The set $\lw_{\sigma}\{I\}=\{\lw_{\sigma}(f)\ |\ f\in I\setminus\{0\}\}\subseteq\langle X\rangle$ is called the *leading word set* of $I$. The two-sided ideal $\lw_{\sigma}(I)=\langle\lw_{\sigma}(f)\ |\ f\in I\setminus\{0\}\rangle\subseteq K\langle X\rangle$ is called the *leading word ideal* of $I$.

-   A subset $G$ of a two-sided ideal $I\subseteq K\langle X\rangle$ is called a *${\sigma}$-Gröbner basis* of $I$ if the set of the leading words $\lw_{\sigma}\{G\}=\{\lw_{\sigma}(f)\ |\ f\in G\setminus\{0\}\}$ generates the leading word ideal $\lw_{\sigma}(I)$.

In the following we focus on computations of Gröbner bases for two-sided ideals in $K\langle X\rangle$. For readers who want to know further properties and applications of non-commutative Gröbner bases, we refer to [@Mo94] and [@Xiu12]. Throughout this paper we assume that ${\sigma}$ is a word ordering on $\langle X\rangle$. The next algorithm is a central part of all Gröbner basis computations.

\[sec2the3\] Let $f\in K\langle X\rangle$, $s\geq 1$, and $G=\{g_{1},\dots,g_{s}\}\subseteq K\langle X\rangle\setminus\{0\}$. Consider the following sequence of instructions.

-   Let $k_{1}=\cdots=k_{s}=0, p=0$, and $v=f$.

-   Find the smallest index $i\in\{1,\dots,s\}$ such that $\lw_{\sigma}(v)=w\lw_{\sigma}(g_{i})w'$ for some words $w,w'\in\langle X\rangle$. If such an $i$ exists, increase $k_{i}$ by 