  For linear non-uniform cellular automata (NUCA) which are local perturbations
of linear CA over a group universe $G$ and a finite-dimensional vector space
alphabet $V$ over an arbitrary field $k$, we investigate their Dedekind
finiteness property, also known as the direct finiteness property, i.e., left
or right invertibility implies invertibility. We say that the group $G$ is
$L^1$-surjunctive, resp. finitely $L^1$-surjunctive, if all such linear NUCA
are automatically surjective whenever they are stably injective, resp. when in
addition $k$ is finite. In parallel, we introduce the ring $D^1(k[G])$ which is
the Cartesian product $k[G] \times (k[G])[G]$ as an additive group but the
multiplication is twisted in the second component. The ring $D^1(k[G])$
contains naturally the group ring $k[G]$ and we obtain a dynamical
characterization of its stable finiteness for every field $k$ in terms of the
finite $L^1$-surjunctivity of the group $G$, which holds for example when $G$
is residually finite or initially subamenable. Our results extend known results
in the case of CA.
