The question on the sheet is: Suppose we have the ring $R=\mathbb{Z} \times \mathbb{Z}_3 \times \mathbb{Z}_5$. Consider the projection map $p:R \to \mathbb{Z}_5$. Given by $p(a,b,c)=c$
1) show that p is a surjective homomorphism
2) Is it unital?
Can someone break down $R=\mathbb{Z} \times \mathbb{Z}_3 \times \mathbb{Z}_5$ looks like? Is it a set of triplets with all the possible combinations of elements in each set? Can someone break down what a projection map is to begin with? What is p, I dont understand how a mapping can be a set with $(a,b,c) = c$, what does that mean? More importantly how does one go about showing in general that something is a surjective homomorphism
Essentially, yes. In general, for a finite product $\prod_{i=1}^n A_i$, you can envision the product as the set of elements $(a_1,\cdots,a_n)$ whereby $a_i \in A_i$ for all $i$. Formally defined,
$$\prod_{i=1}^n A_i = \{ (a_1,a_2,\cdots,a_n) \mid a_i \in A_i \}$$
A more familiar example of this type of product comes from the Cartesian plane (the $xy$ plane) or Cartesian $3D$ space, which are $\Bbb R^2, \Bbb R^3$ respectively. ($\Bbb R^2 = \Bbb R \times \Bbb R$, and similar for the cube.) Then, for example,
$$\Bbb R^2 = \Bbb R \times \Bbb R = \{ (x,y) \mid x,y \in \Bbb R \}$$
Of course, this definition is usually introduced with sets - as opposed to groups/rings/fields/etc. - but all that really changes is that now that Cartesian product is associated with an operation now in those cases, for example.
This is another thing that is convenient to first think of in terms of familiar, Cartesian geometry. Consider $\Bbb R^2$ again. A projection would be defined as a function $\Bbb R^2 \to \Bbb R$ which is surjective, of course, and meets all of the other associated properties. You could define this map by the definition $(x,y) \mapsto x$. What this does, in a heuristic/informal/visual manner, is take all of the points from the $xy$ plane, and "drag" them straight up or down to the $x$ axis. You could similarly have another projection defined by $(x,y) \mapsto y$, to drag each point horizontally to the $y$ axis.
Perhaps consider instead $\Bbb R \times \Bbb Z$, which is the set of points $(x,y)$ such that $x$ is real and $y$ is an integer. This, visually, is like a bunch of horizontal stripes along the $xy$ plane, almost like the $x$ axis was duplicated infinitely many times. Each "stripe" is at some integer value $y$ and goes infinitely in any direction. In this example, we can use the same projection definitions as before. If $(x,y) \mapsto x$, then, again, all of the "stripes" are dragged down to merge with the $x$ axis. If $(x,y) \mapsto y$, then each "stripe" collapses in on itself, becoming the single integer point on the $y$ axis it stems from.
What you're beginning to deal with are these same notions, but in a more abstract, and often more difficult to visualize, environment. Essentially, a projection is a function $p : \prod A_i \to A_k$ that sends all of the elements of a product to a single set from the product, "collapsing" it in some sense, and each element of the product is sent to its "original" element from the single set. (By this I mean, for example, $(x,y) \mapsto x$ in $\Bbb R^2$ can send $(2,6)$ to $2$, the element that pair is associated with in $\Bbb R$. Of course this can often be a noninjective function since, in the same scenario, $(2,7)$ and $(2,8)$, or, really, any $(2,y)$ is sent to $2$. They're all associated with $2$ and thus sent there.)
The definition of this more general projection $p : \prod A_i \to A_k$ will typically be of the form $(a_1, \cdots, a_n) \mapsto a_k$ (where $k$ is some fixed number; $k$ might be $3$ for example, and refer to $A_3$, the third set in the product).
I think Captain Lama address this point sufficiently in their post, but to be clear: projections are (typically) multi-valued functions. Remember from multivariable calculus how you might have functions like $f(x,y) = x^2 + y^2$ or $g(x,y,z) = \sin(z)\cos(y)\tan(x)$ or whatever -- functions that took in more than one value?
It's similar here. Those functions, when framed differently, took in a "tuple" of elements. If you wanted a function that took in inputs from the $xy$ plane, you'd use $f(x,y)$ as your definition - since you're taking in a pair $(x,y)$ and using $f$ to generate an output. It was just always understood back then that $x,y,z,$ and so on, were real numbers. That is, if you have just $x,y$, you're working with $\Bbb R^2$ for instance.
Your projections also take in a tuple of elements, and output an element from a single set, so it's a similar idea. It's just now you might be dealing with tuples whose elements are not all from the same set.
It's really as simple as that. Show it has the properties of a homomorphism that you expect, and show that every element of the codomain has a preimage.