Abstract We introduce the notion of radical preservation and prove that a radical‐preserving homomorphism of left artinian rings of finite projective dimension with superfluous kernel reflects the finiteness of the little finitistic, big finitistic, and global dimension. As an application, we prove that every bound quiver algebra with quasi‐uniform Loewy length, a class of algebras introduced in this paper, has finite finitistic dimensions. The same result holds more generally in the context of semiprimary rings. Moreover, we construct an explicit family of such finite‐dimensional algebras where the finiteness of their big finitistic dimension does not follow from existing results in the literature.