Does the notation $x \mapsto(x+2 \mathbb{Z}, x+3 \mathbb{Z})$ refers to $x \mapsto(x+2 k, x+3 k) , k \in Z$ I found a notation in a problem and which was not used in the lecture and I didn't cross before:
It asks to show that the application is a ring homomorphism:
$$
\begin{aligned}
\varphi: \mathbb{Z} & \longrightarrow \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 3 \mathbb{Z} \\
& x \quad \mapsto(x+2 \mathbb{Z}, x+3 \mathbb{Z})
\end{aligned}
$$
I looked up wikipedia pages on "ring" and modular arithmetic but I don't find use of such notation, I am guessing it is some convention of notation and it refers to $x \mapsto(x+2 k, x+3 k) , k \in Z$ am I correct?
 A: No, your interpretation is not correct.
$x + 2\mathbb{Z}$ is the coset of $\mathbb{Z}/2\mathbb{Z}$ that contains $x$. So for example if you write the elements of $\mathbb{Z}_n$ as $\{0, 1, \ldots, n-1\}$ then in this example the image of $8$ is the pair $(0,2)$.
A: This is part of a somewhat broader bit of notation for quotient rings such as $\mathbb Z/2\mathbb Z$. In particular, $2\mathbb Z$ refers literally to the set of even numbers $\{2k : k\in \mathbb Z\}$ and $1+2\mathbb Z$ refers to the set of odd numbers $\{1+2k:k\in \mathbb Z\}$. It's fairly common to see arithmetic done on sets this way, where we use a set like $\mathbb Z$ in an expression to stand in for "the set of all possible evaluations of this expression" - for instance, one sometimes sees equations even like $4\mathbb Z + 6 \mathbb Z = 2\mathbb Z$, which is literally true of sets. This is related to various operations on ideals in ring theory (although one should be really careful that it is far less common to use this notation with sets being multiplied together - for the good reason that multiplication in a quotient ring is not necessarily multiplication of each element of a coset).
The point of the notation is that the ring $\mathbb Z/2\mathbb Z$ consists of the additive cosets of $2\mathbb Z$ - it is a ring whose two elements are "the set of even numbers" and "the set of odd numbers" and the usual projection that reduces mod $2$ happens to be given by
$$f:\mathbb Z\rightarrow\mathbb Z/2\mathbb Z$$
$$f(x)=x+2\mathbb Z.$$
It's common to represent the elements of $\mathbb Z/2\mathbb Z$ by the numerals $\{0,1\}$ - or sometimes with various ornaments like $\bar 0$ and $\bar 1$ which tend to be defined as "the set of even numbers $2\mathbb Z$" or "the set of odd numbers $1+2\mathbb Z$", but the literal meaning of a quotient involves these cosets - which are given by shifting the set $2\mathbb Z$ by addition. Note that the notation of cosets is rather flexible, since it lets you write $0+2\mathbb Z$ and $2+2\mathbb Z$ and $4+2\mathbb Z$ and so on as equally valid (and equal) elements of $\mathbb Z/2\mathbb Z$. Mod $3$ works similarly.
So the notation
$$g(x)=(x+2\mathbb Z,x+3\mathbb Z)$$
is really sending each $x\in \mathbb Z$ to a pair consisting of a coset of $2\mathbb Z$ (in $\mathbb Z/2\mathbb Z$) and a coset of $3\mathbb Z$ - as is required by the literal definition of these things. That is, it takes an integer to a pair of sets, the first set being in $\mathbb Z/2\mathbb Z$, the second in $\mathbb Z/3\mathbb Z$.
