# Expressing a quotient of $\mathbb{Z}/N$ with $N$ a $\mathbb{Z}$-module as the Smith Normal Form

Consider the $\mathbb{Z}$-module $M=\mathbb{Z}/N$. I have to express $M$ as the usual (with $a_i|a_{i+1}$ and so on): $$(\mathbb{Z}/(a_1))\oplus(\mathbb{Z}/(a_2))\oplus...\oplus(\mathbb{Z}/(a_s))\oplus\mathbb{Z}^t$$ Where $N\in \mathbb{Z}^2$ the $\mathbb{Z}$-module:

1) $N=<(0,5)^t>$

2) $N=<(2,1)^t>$

3) $N=<(4,2)^t,(6,3)^t>$

The thing is that I don't think I'm interpreting this well. In Artin's Algebra book what I've done is, for example with $N_2$: $<(2,1)^t>\rightarrow <(0,1)^t>$ so $N\cong \mathbb{Z}/(0)\oplus\mathbb{Z}/(1) \cong \mathbb{Z}$ And then I'll just do $$M = \mathbb{Z}/N \cong \mathbb{Z}/\mathbb{Z} \cong \{0\}$$ It's this correct? Or what's the thing i don't understand? If so, how can you do 1)? I get this: $$M = \mathbb{Z}/N \cong \mathbb{Z}/(\mathbb{Z}\oplus\mathbb{Z}/5)$$ And I'm stuck there. Any help?

• Here are some related posts on this topic: 1, 2, 3, 4. Do any of these help? – André 3000 Jan 13 '18 at 1:28
• Yes! Thanks a lot – Rafa Jan 13 '18 at 1:31