Let ${\Bbb Z}I$ be the free abelian group with basis $I$, let $\chi$ be a pair of integral bilinear forms on ${\Bbb Z}I$. We will endow the free $K$-algebra $K\langle I\rangle$ generated by $I$ with a comultiplication which depends on $\chi$. This yields an associated bilinear form on $K\langle I\rangle$ which may be called the Drinfeld form. We are going to show that certain elements of $K\langle I\rangle$ which are similar to the well-known quantum Serre relations belong to the left radical of the Drinfeld form, provided certain integrality conditions are satisfied.