Let $k$ be a field. A polynomial of the form $l=a_1x_1+\cdots+a_nx_n$ is called a linear form ($a_i\in k$), and its support is the set of all variables $x_i$ such that $a_i\neq 0$.
Let $L\subseteq k[x_1,\ldots,x_n]$ be an ideal generated by some linear forms. Let $S$ be the set of all linear forms in $L$ whose support is minimal (i.e., there doesn't exist another nonzero linear form with the support being a strict subset of this set.)
I can show that $S$ is finite, and I want to show that $S$ is a Grobner basis of $L$ by using Buchberger's criterion. So, suppose $A=x_k+\cdots,B=x_l+\cdots$ are two elements of $S$ (where the leading coefficient is scaled to be $1$. If $k\neq l$, I want to show that $x_lA-x_kB$ leaves zero remainder when dividing by elements of $S$. How to show that?