Tensor Product

We must show that every element of $F\otimes M$ is an $F$-linear combination of elements of $\{1\otimes x \}$. But elements of $F\otimes M$ are of the form $\sum_{i\le k} a_i\otimes m_i$ for some $a_i \in F$ and $m_i \in M$. So it suffices to show that each $a\otimes m$ is an $F$-linear combination of elements of $\{1\otimes x \}$. But $X$ generates $M$ so we have $m=\sum_{i\le n} r_i x_i$ for some $r_i \in R$ and $x_i \in X$. Now $$ a\otimes m=a\otimes (\sum r_i x_i )= \sum a\otimes r_i x_i =\sum ar_i \otimes x_i =\sum ar_i (1\otimes x_i )$$Note that $ar_i \in F$. So each $a\otimes m$ is an $F$-linear combination of elements of $\{1\otimes x \}$, and thus $$ \sum_{i\le k} a_i\otimes m_i=\sum_{i\le k}\sum _{j\le n_i} a_{ij} (1\otimes x_{ij})$$is also an $F$-linear combination of elements of $\{1\otimes x \}$.