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We show that the image of a subshift $X$ under various injective morphisms ofsymbolic algebraic varieties over monoid universes with algebraic varietyalphabets is a subshift of finite type, resp. a sofic subshift, if and only ifso is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ beArtinian modules over a ring. We prove that for every closed subshift submodule$\Sigma \subset A^G$ and every injective $G$-equivariant uniformly continuousmodule 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 overadmissible Artinian group structure alphabets are also obtained.

On images of subshifts under injective morphisms of symbolic varieties