Let $R$ be a ring, $M$ an $R$-module and $S=\left\{m_{1}, \ldots, m_{k}\right\}$ a subset of $M$. Define what it means to say $S$ spans $M$. Define what it means to say $S$ is an independent set.

We say $S$ is a basis for $M$ if $S$ spans $M$ and $S$ is an independent set. Prove that the following two statements are equivalent.

1. $S$ is a basis for $M$.

2. Every element of $M$ is uniquely expressible in the form $r_{1} m_{1}+\cdots+r_{k} m_{k}$ for some $r_{1}, \ldots, r_{k} \in R$.

We say $S$ generates $M$ freely if $S$ spans $M$ and any map $\Phi: S \rightarrow N$, where $N$ is an $R$-module, can be extended to an $R$-module homomorphism $\Theta: M \rightarrow N$. Prove that $S$ generates $M$ freely if and only if $S$ is a basis for $M$.

Let $M$ be an $R$-module. Are the following statements true or false? Give reasons.

(i) If $S$ spans $M$ then $S$ necessarily contains an independent spanning set for $M$.

(ii) If $S$ is an independent subset of $M$ then $S$ can always be extended to a basis for $M$.