Are all unital endormorphisms of a Weyl algebra automorphisms? Given $k$ a field of characteristic $0$, let $A_n(k)$ be the $n$-th Weyl algebra over $k$ – i.e. the unital algebra generated by elements $p_1, ..., p_n$ and $q_1, ..., q_n$ modulo the relations $[p_i, q_i]=1$ and $[p_i, p_j]=[q_i, q_j]=[p_i, q_j]=0$ for all $i\neq j$. Then the Dixmier conjecture states that every endomorphism of $A_n(k)$ is an automorphism.
Hence let $\eta:A_n(k)\rightarrow A_n(k)$ be an algebra homomorphism. By an induction argument it can be shown that $A_n(k)$ is a simple algebra, and so $\eta$ must be injective. Now, if $\eta$ is unital, then for each $i$ we must have $[\eta(p_i), \eta(q_i)]=\eta([p_i, q_i])$=1. It is a routine computation to show that the commutator of two elements of $A_n(k)$ is $1$ iff those two elements are $p_j$ and $q_j$ for some $j$, for instance by considering the canonical decomposition of each element of $A_n(k)$ into a sum of monomial terms where all $p_i$s are to the left of all $q_i$s; hence $\eta$ must map each pair of non-commuting generators to another pair of non-commuting generators. By injectivity the images of each of these pairs must be distinct, and so in particular $\{p_1, ..., p_n, q_1, ..., q_n\}\subset Im(\eta)$. Surjectivity follows immediately. Is this argument correct?
The only reason I ask is that I've never come across non-unital homomorphisms of a unital ring besides the zero homomorphism; what are some other contexts in which these are considered?


It is a routine computation to show that the commutator of two elements of $A_n(k)$ is $1$ iff those two elements are $p_j$ and $q_j$ for some $j$, for instance by considering the canonical decomposition of each element of $A_n(k)$ into a sum of monomial terms where all $p_i$s are to the left of all $q_i$s

This is wrong.  For instance, $[cp_j,c^{-1}q_j]=1$ for any nonzero $c\in k$.  Or some less trivial examples: $$[p_1+p_2+p_3,q_1-q_2+q_3]=1$$ and $$[p_1,q_1+p_2q_5^3+p_1^2]=1.$$
