Special linear group over a quotient of a polynomial ring 
I'm interested in the structure of groups of the form $SL_2(A)$ where $A$ is a ring of the form $A=\mathbb{F}_p[x]/(x^2)$, or more generally $A=\mathbb{F}_p[x]/(x^n)$.

It is clear that for $n=1$ we just recover $SL_2(\mathbb{F}_p)$ which is well-understood. Also if $p(x) \in \mathbb{F}_p[x]$ is irreducible then $SL_2(\mathbb{F}_p[x]/(p(x))) = SL_2(\mathbb{F}_q)$ for some finite field $\mathbb{F}_q$, and again there exists a vast literature on these groups. My question is whether special linear groups over these slightly different (local) rings $\mathbb{F}_p[x]/(x^n)$ have been studied at all, and in particular their subgroups structure. Any references would be greatly appreciated.
 A: I don't know what you mean by subgroup structure, but here is a couple of things we can say about these groups.
Set $A_n=\mathbb{F}_p[x]/(x^n)$.

*

*the group $SL_2(A_n)$ is generated by  transvection matrices.

Sketch of proof. Lift an element of $\bar{M}\in SL_2(A_n)$ to an element of $M\in M_2(\mathbb{F}_p[x])$. Then there are two matrices $U,V\in GL_2(\mathbb{F}_p[x])$ which are both product of transvection matrices of $M_2(\mathbb{F}_p[x])$ such that $UMV=diag(P_1,P_2), P_i\in \mathbb{F}_p[X]$ (we may even ask $P_1\mid P_2$, but we don't need it here)
Now it is known (and true over any ring) that there are two matrices  $U',V'\in GL_2(\mathbb{F}_p[x])$ which are both product of transvection matrices of $M_2(\mathbb{F}_p[x])$ such that $U' diag(P_1,P_2)V'=diag (1,P_1P_2)$.
All in all, there are  $U'',V''\in GL_2(\mathbb{F}_p[x])$ which are both product of transvection matrices of $M_2(\mathbb{F}_p[x])$ such that $U'' MV''=diag (1,P_1P_2)$.
Note that $\det(M)=P_1P_2$, so $\overline{P_1}\ \overline{P_n}=\overline{\det(M)}=\det(\overline{M})=\overline{1}$. Thus $\overline{U''} \ \overline{M} \ \overline{V''}=I_2$. Since the class of a transvection matrix of $SL_2(\mathbb{F}_p[x])$ is a transvection matrix of $SL_2(A_n)$, we are done, taking into account that the inverse of a transvection matrix is a transvection matrix.

*

*the group $SL_2(A_n)$ is generated by transvection matrices $T_{12}(\bar{P}), \bar{P}\in A_n$  and $\begin{pmatrix}\bar{0} & \bar{1} \cr -\bar{1} & \bar{0}\end{pmatrix}$
Proof. follows from the previous point and the relation $ST_{12}S^{-1}=T_{21}.$

*

*$SL_2(A_2)=L\rtimes SL_2(\mathbb{F}_p)$ , where $L=\{M\in M_2(\mathbb{F}_p)\mid tr(M)=0\}$ and $SL_2(\mathbb{F}_p)$ acts by conjugation on $L$.

Sketch of proof. Indeed, the reduction morphism $SL_2(A_2)\to SL_2(\mathbb{F}_p)$ is surjective (the image contains any transvection matrix), with kernel $K=\{I_2+\bar{x} M, M\in L\}$. This morphism obviously splits.
Now $M\in L\mapsto I_2+\bar{x} M\in K$ is easily seen to be a group isomorphism (since $\bar{x}^2=\bar{0}$, this map will be a group morphism).
Note that $L\simeq \mathbb{F}_p^3$, but the action of $SL_2(\mathbb{F}_p$ is easier to describe using $L$.
I don't know if it is the kind of results you were looking for.
