Let $\mathbb{F}_p$ be a finite field for an odd prime $p$. Consider the ring $$\mathcal{L}(\mathbb{F}_p(X),Y)$$ of additive (or linearized) polynomials in $Y$ over the rational function field $\mathbb{F}_p(X)$. That is, we are interested in polynomials of the form $$a_n(X)Y^{p^n} + a_{n-1}(X)Y^{p^{n-1}} + \dots + a_1(X)Y^{p} + a_0(X)Y$$ with $a_j(X) \in \mathbb{F}_p(X)$ for every $0 \leq j \leq n$. Addition is the usual, while multiplication is now function composition, making $\mathcal{L}(\mathbb{F}_p(X),Y)$ a non-commutative ring with unity.
My interest is regarding this non-commutativity of the multiplication in $\mathcal{L}(\mathbb{F}_p(X),Y)$ and its structure as a ring.
What can we say about the structure of the multiplicative monoid of $\mathcal{L}(\mathbb{F}_p(X),Y)$?
What is the center of $\mathcal{L}(\mathbb{F}_p(X),Y)$? It is clear that the center contains $\mathbb{F}_p Y$, but is that the whole center?
The same questions can be asked of the ring $\mathcal{L}(\mathbb{F}_p[X],Y)$ of linearized polynomials with coefficients now from the domain $\mathbb{F}_p[X]$ instead of the rational function field.
What can we say about the structure of the multiplicative monoid of $\mathcal{L}(\mathbb{F}_p[X],Y)$?
What is the center of $\mathcal{L}(\mathbb{F}_p[X],Y)$? Again, it is clear that the center contains $\mathbb{F}_p Y$, but is that the whole center?
This is only the surface. Even more interesting is how this multiplication behaves with regard to absolute irreducibility of the polynomials, its roots (in an extension of $\mathbb{F}_p(X)$), its splitting fields, etc. But I'll get to those later.
Apologies if the question is broad, and will refine it as I get more familiar with the object . I am interested in all aspects of the structure of the non-commutative multiplication here, so any comprehensive references would be valuable. The ones I googled seem way too sophisticated and abstract (Drinfeld modules and the like) that I don't know where to even begin.
Please feel free to share any interesting facts about the non-commutative ring $\mathcal{L}(\mathbb{F}_p[X],Y)$ (and its multiplicative monoid) that you know of. Thanks.