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In this note, we prove that if is an Artin algebra with a simple module of finite projective dimension, then the finiteness of the finitistic dimension of implies that of where is the primitive idempotent supporting . We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if is the quotient of a path algebra by an admissible ideal whose defining relations do not involve a certain arrow , then the finitistic dimension of is finite if and only if the finitistic dimension of is finite. References