Let G be a group. Let X be a connected algebraic group over an algebraically closed field K. Denote by A=X(K) the set of K-points of X. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over (G,X,K). They are cellular automata τ:AG→AG whose local defining map is induced by a homomorphism of algebraic groups XM→X where M⊂G is a finite memory set of τ. Our first result is that when G is sofic, such an algebraic group cellular automaton τ is invertible whenever it is injective and char(K)=0. When G is amenable, we show that an algebraic group cellular automaton τ is surjective if and only if it satisfies a weak form of pre-injectivity called ▪-pre-injectivity. This yields an analogue of the classical Moore-Myhill Garden of Eden theorem. We also introduce the near ring R(K,G) which is K[Xg:g∈G] as an additive group but the multiplication is induced by the group law of G. The near ring R(K,G) contains naturally the group ring K[G] and we extend Kaplansky's conjectures to this new setting. Among other results, we prove that when G is an orderable group, then all one-sided invertible elements of R(K,G) are trivial, i.e., of the form aXg+b for some g∈G, a∈K⁎, and b∈K. This in turns allows us to show that when G is locally residually finite and orderable (e.g. Zd or a free group), all injective algebraic cellular automata τ:CG→CG are of the form τ(x)(h)=ax(g−1h)+b for all x∈CG,h∈G for some g∈G, a∈C⁎, and b∈C.