6. (i) Show that $N=\{12, 14, 21\}$ generates $\mathbb{Z} $ as a $\mathbb{Z} $-module. [2 Marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 Marks] generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. (iii) For a $\mathbb{Z}$-module $A$, denote by $T(A)$ the torsion subgroup of $A$. (a) Find the elements in $T(\mathbb{R} / \mathbb{Z})$. [3 Marks] (b) Let $P, Q$ be $\mathbb{Z}$-modules and $\phi: P ightarrow $ a $\mathbb{Z}$-module map. Show that $\phi(TP) \subseteq TQ$. [3 Marks] (iv) Let $R$ be a commutative ring with identity, and $m \in R$ be a fixed element. Show that $\phi: R ightarrow R \cdot m$ defined by $\phi(r)=- \cdot m$ is a left $R$-module homomorphism, where $R \cdot m=\{r \cdot m: r \in R\}$. [5 Marks ] CS.VS. 1470