Let $A$ be a unital algebra and $d_u$ is universal differential given by $d_u(x)=1 \otimes x-x \otimes 1$. It is viewed as the map $A \to \Omega_u^1(A)$ where $\Omega_u^1(A)=\ker m$ is the kernel of multiplication map $A \otimes A \to A$. One shows that $\Omega_u^1(A)$ is generated (as a left $A$ module) by $d_u(A)$ and since $d_u(1)=0$ the map $A / (\mathbb{C}1) \ni \overline{y} \mapsto d_u(y)$ is well defined. Consider the mapping $x \otimes \overline{y} \mapsto xd_u(y)$ acting as $A \otimes A /\mathbb{C}1 \to \Omega_u^1(A)$.
Question Why this map is one-to-one?