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Split-by-nilpotent extensions algebras and stratifying systems

Let $Γ$ and $Λ$ be artin algebras such that $Γ$ is a split-by-nilpotent extension of $Λ$ by a two sided ideal $I$ of $Γ.$ Consider the so-called change of rings functors $G:={}_ΓΓ_Λ\otimes_Λ-$ and $F:={}_ΛΛ_Γ\otimes_Γ-.$ In this paper, we find the necessary and sufficient conditions under which a stratifying system $(Θ,\leq)$ in $\moduΛ$ can be lifted to a stratifying system $(GΘ,\leq)$ in $\modu\,(Γ).$ Furthermore, by using the functors $F$ and $G,$ we study the relationship between their filtered categories of modules and some connections with their corresponding standardly stratified algebras are stated. Finally, a sufficient condition is given for stratifying systems in $\modu\,(Γ)$ in such a way that they can be restricted, through the functor $F,$ to stratifying systems in $\modu\,(Λ).$