  In 2010, V. Futorny and S. Ovsienko gave a realization of
$U(\mathfrak{gl}_n)$ as a subalgebra of the ring of invariants of a certain
noncommutative ring with respect to the action of $S_1\times
S_2\times\cdots\times S_n$, where $S_j$ is the symmetric group on $j$
variables. An interesting question is what a similar algebra would be in the
invariant ring with respect to a product of alternating groups. In this paper
we define such an algebra, denoted $\mathscr{A}(\mathfrak{gl}_n)$, and show
that it is a Galois ring. For $n=2$, we show that it is a generalized Weyl
algebra, and for $n=3$ provide generators and a list of verified relations. We
also discuss some techniques to construct Galois orders from Galois rings.
Additionally, we study categories of finite-dimensional modules and generic
Gelfand-Tsetlin modules over $\mathscr{A}(\mathfrak{gl}_n)$. Finally, we
discuss connections between the Gelfand-Kirillov Conjecture,
$\mathscr{A}(\mathfrak{gl}_n)$, and the positive solution to Noether's problem
for the alternating group.
