This paper, authored by Hyman Bass, explores the concept of finitistic dimension in ring theory, building upon the work of Kaplansky and others. The main focus is on the study of commutative Noetherian rings and their properties, particularly the condition where the projective dimension of every module is either 0 or infinite. The paper introduces the notion of a "T-nilpotent" ideal and characterizes commutative rings with this property as direct sums of finite numbers of local rings, each with a T-nilpotent maximal ideal. The author then generalizes these results to noncommutative rings, proving a theorem that characterizes rings for which the finitistic global dimension vanishes. This theorem is a noncommutative extension of Kaplansky's original result. The paper also discusses the relationship between projective covers, injective envelopes, and perfect rings, providing detailed proofs and examples to illustrate the concepts. Key results include: 1. The equivalence of conditions for a ring to be left perfect, including the requirement that the Jacobson radical is T-nilpotent. 2. The characterization of rings with zero finitistic projective dimension in terms of their modules and ideals. 3. The proof that if a ring is left perfect and every finitely generated proper right ideal has a nonzero left annihilator, then the finitistic projective dimension is zero. The paper concludes with a discussion on the finitistic injective dimension and its implications for injective modules and ideals.This paper, authored by Hyman Bass, explores the concept of finitistic dimension in ring theory, building upon the work of Kaplansky and others. The main focus is on the study of commutative Noetherian rings and their properties, particularly the condition where the projective dimension of every module is either 0 or infinite. The paper introduces the notion of a "T-nilpotent" ideal and characterizes commutative rings with this property as direct sums of finite numbers of local rings, each with a T-nilpotent maximal ideal. The author then generalizes these results to noncommutative rings, proving a theorem that characterizes rings for which the finitistic global dimension vanishes. This theorem is a noncommutative extension of Kaplansky's original result. The paper also discusses the relationship between projective covers, injective envelopes, and perfect rings, providing detailed proofs and examples to illustrate the concepts. Key results include: 1. The equivalence of conditions for a ring to be left perfect, including the requirement that the Jacobson radical is T-nilpotent. 2. The characterization of rings with zero finitistic projective dimension in terms of their modules and ideals. 3. The proof that if a ring is left perfect and every finitely generated proper right ideal has a nonzero left annihilator, then the finitistic projective dimension is zero. The paper concludes with a discussion on the finitistic injective dimension and its implications for injective modules and ideals.