On split-by-nilpotent extensions
Let $A$ and $R$ be two artin algebras such that $R$ is a split extension of $A$ by a nilpotent ideal. We prove that if $R$ is quasi-tilted, or tame and tilted, then so is $A$. Moreover, generalizations of these properties, such as laura and shod, are also inherited. We also study the relationship between the tilting $R$-modules and the tilting $A$-modules.