  We show that the image of a subshift $X$ under various injective morphisms of
symbolic algebraic varieties over monoid universes with algebraic variety
alphabets is a subshift of finite type, resp. a sofic subshift, if and only if
so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be
Artinian modules over a ring. We prove that for every closed subshift submodule
$\Sigma \subset A^G$ and every injective $G$-equivariant uniformly continuous
module homomorphism $\tau \colon \Sigma \to B^G$, a subshift $\Delta \subset
\Sigma$ is of finite type, resp. sofic, if and only if so is the image
$\tau(\Delta)$. Generalizations for admissible group cellular automata over
admissible Artinian group structure alphabets are also obtained.
