Facts about Weyl algebra

Question

I am trying to prove a few things about the first Weyl algebra, $W = k[x,y]/(xy-yx-1)$ over an algebraically closed field $k$ with $char(k)=p>0$. In particular, I am interested in nilpotent elements.
First, it is known that the center is precisely $Z(W) = k[x^p,y^p]$, generated by $x^p,y^p$. Also, the quotient $W/Z(W)$ is a free module of rank $p^2$, since a basis is given by $\{x^iy^j \mid 0 \le i,j \le p-1 \}$. Another known fact is that any irreducible representation of $W$ (in char $p$) must be of dimension $p$.
Now suppose we are interested in nilpotent elements, so let $R$ be a homomorphic image of $W$ such that $x,y$ are nilpotent, or equivalently let $R=W/I$ for some ideal such that $x,y$ are nilpotent modulo $I$.
Is it true that $R \cong M_p (Z(W))$?
Consider the center $Z(W)$. This is a finite dimensional commutative (Artinian) algebra, so it must a direct product of local rings $$Z(W) \cong Z_1 \times \dots \times Z_m,$$ where each $Z_i$ is a local commutative ring.
Then modulo the Jacobson radical, do we have $R/Jac(R) \cong W/(x^p,y^p) \cong M_p(k)$?
By this, $R$ has a unique irreducible module $S$ (up to isomorphism) and $\dim(S) = p$.
Then can we say $R \cong P \oplus \dots \oplus P$, where $P$ is projective indecomposable with $P/Jac(P) \cong S$ and $P$ unique up to isomorphism?
Then $R/Jac(R)$ would be a direct sum of $p$ copies of $S$, so $R$ would be a direct sum of $p$ copies of $P$.

Answer 1

It is true that $W/(x^p,y^p)\cong M_p(k)$. In fact, we have a more general result. For every $a,b\in k$ the algebra $W/(x^p-a,y^p-b)$ is a CSA (central simple algebra) of rank $p^2$. If, $b=0$ this CSA is trivial, i.e. it is $\cong M_p(k)$. If $b\neq 0$, then the image of this CSA in the Brauer group is $[ab,b)$.

Here $[\cdot,\cdot ):k/\wp (k)\times k^\times/k^{\times 2}\to{\rm Br(k)}$ is the Artin-Schreier symbol. (And $\wp :k\to k$ is the Artin-Schreier morphism, given by $x\mapsto x^p-x$.)

Unfortunately, I don't know any references for this.