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 $\phi: P ightarrow Q$ 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 $\phi(r)=r \cdot m$ is a left $R$-module homomorphism, where SR \cdot m=\{r \cdot m: r \in R\}$. [5 Marks] CS.VS. 1466