Corollary 4

Let A be a ring with no divisor of zero, E a free right A-module and F a free left A-module. Then the relation x $\otimes$ y = 0 in E $\otimes_A$ F implies x = 0 or y = 0.

Let $(a_\lambda)$ be a basis of E, $(b_\mu)$ a basis of F and let $x = \sum_\lambda a_\lambda \xi_\lambda, y = \sum_\mu \eta_\mu b_\mu$; then $x \otimes y = \sum_{\lambda, \mu} ((a_\lambda \xi_\lambda \eta_\mu) \otimes b_\mu$ and the relation $x \otimes y = 0$ implies $\xi_\lambda \eta_\mu = 0$ for every ordered pairs of indices $(\lambda, \mu)$ (Corollary 1). Hence, if $x \neq 0$, that is $\xi_\lambda \neq 0$ for at least one $\lambda$, it follows that $\eta_\mu = 0$ for all $\mu$, whence $y = 0$.