JOURNAL OF ALGEBRA 11, 483-487 (1969) A Note on Homological Dimension* JOEL NI. COHEN Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 and University of Chicago, Chicago, Illinois 60637 Communicated by I. N. Herstein Received May 8, 1968 The purpose of this note is to answer some questions concerning the follow- ing problem: Let I be a twosided projective ideal of a ring R. Let R* = R/I. Then any R*-module iz is an R-module and d,A < d,*A + 1. When do we have equality ? First we recall that a proof of the fact that d,A < d,t.4 + 1 (cf. for example [5], Theorem 5.2) makes it clear that if equality holds when d,*A< 1 then equality holds whenever d,*A < a3. (In this case, however, it is still possible to have d,*A = co and d,A < co. For example, R = 2, I = 42, A = 2, .) This will be precisely our method of proof, hence we shall not consider the case where d,aA = CD. The dimension d,A will always refer to the left homological dimension: i.e. the minimum length of a left projective resolution of A. Thus d,iz = 0 if and only if A is a nonzero projective. dRO x --c/3. All rings have identity. We shall make use of the following lemma: LEMMA. If M is a left jut R-module and I C R is a right ideal, then the action p : I (2~~ M-+ IM is an isomorphism. Proof. To show that p is a monomorphism, tensor the inclusion It-t R with M which is left flat yielding the inclusion I OR MC-+ R OR M. Com- posing with the isomorphism R OR M NM yields I @a MC+ M with image IM by definition. COROLLARY. If I # I2 is a proper twosided ideal of R then R/I is not (right or left) R-jlat. * This work was partially supported by NSF grant GP-5609. 483