Vol. 15, No. 1, 1965

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Finitistic global dimension for rings

Horace Yomishi Mochizuki

Vol. 15 (1965), No. 1, 249–258
Abstract

The finitistic global dimensions lfPD(R), lFPD(R), and lFID(R) are defined for a ring R. We obtain the following results for R semiprimary with Jacobson radical N. C is a simple left R-module and l.dimRC < , and suppose that l.dimRNi1Ni < for i 3. Then m lfPD(R) = lFPD(R) (m + 1). Theorem 2: Suppose that l.inj.dimRP l.inj.dimRRN2 < for every projective (RN2)-module P and that l.inj.dimRNi1 Ni < for i 3. Then lFID(R) = l.inj.dimRR < . The method of proof uses a result of Eilenberg and a result of Bass on direct limits of modules together with the lemma: If M is a left R-module such that Nk1M0 and NkM = 0, then every simple direct summand of Nk1 is isomorphic to a direct summand of Nk1Nk.

Mathematical Subject Classification
Primary: 16.90
Milestones
Received: 17 January 1964
Published: 1 March 1965
Authors
Horace Yomishi Mochizuki