The tensor product of modules $M_0, M_1$ is a quotient of a free module $F$, …, by a submodule F'. I found 2 definitions of this F', and the difference is in this generating rules:
- Wikipedia: $(r\cdot m_0)\otimes m_1 - m_0\otimes (r\cdot m_1)$;
- Lang's “Algebra”: $r\cdot(m_0\otimes m_1) - (r\cdot m_0)\otimes m_1, r\cdot(m_0\otimes m_1) - m_0\otimes (r\cdot m_1)$.
IMO Lang's “Algebra” $\to$ Wikipedia, because
$(r\cdot(m_0\otimes m_1) - m_0\otimes (r\cdot m_1)) - (r\cdot(m_0\otimes m_1) - (r\cdot m_0)\otimes m_1)$ $= (r\cdot m_0)\otimes m_1 - m_0\otimes (r\cdot m_1)$.
Does the backward implication hold? If not, then it is an error in Wikipedia?