On images of subshifts under injective morphisms of symbolic varieties

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 $\mathrm{\Sigma }\subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $\tau \text{\hspace{0.17em}⁣}:\mathrm{\Sigma }\to B^G$, a subshift $\Delta \subset \Sigma$isoffinitetype,resp\mathrm{.}sofic,ifandonlyifsoistheimage$\tau(\Delta)$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.