Describe $\mathbb{Z} \times \mathbb{Z} /\mathbb{3Z}\times\mathbb{Z} $

Question

I am trying to describe this quotient group $\mathbb{Z}\times\mathbb{Z}/\mathbb{3Z}\times\mathbb{Z}$ Let's denote with $A$ and $B$ respectively $\mathbb{Z} \times \mathbb{Z} $ and $\mathbb{3Z}\times\mathbb{Z}$

$A / B:= \{a+B : a \in A\}$ my problem is (might be a stupid one) there are infinity of $a \in A$ in form $(a_1,a_2)$ so i am confused about how to built that quotient group.

Answer 1

@lhf has given the right solution, but your comment suggests you don't understand it.

Here's a more elementary way.

We want to give a list of the distinct cosets $a+B$ - I use your notation.

When does $a+B=x+B$?. Answer, if and only if $a-x\in B$.

Taking $a=(m,n)$ and $x=(u,v)$ we have that these cosets are the same if and only if$(m-u,n-v)\in B=3\mathbb{Z}\times \mathbb{Z}$.

That is $m=u$ is a multiple of $3$ and $n-v$ is any integer.

So for a set of cosets we can take the following set of 3 cosets : $\{(0,0)+B, (1,0)+B, (2,0)+B\}$.

Answer 2

Hint: Consider $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}/3\mathbb{Z}$ given by $(a,b) \mapsto a \bmod 3$. Find its kernel and its image.