#### 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 $lfPD\left(R\right)$, $lFPD\left(R\right)$, and $lFID\left(R\right)$ 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 $l.{dim}_{R}C<\infty$, and suppose that $l.{dim}_{R}{N}^{i-1}∕{N}^{i}<\infty$ for $i\geqq 3$. Then $m\leqq lfPD\left(R\right)=lFPD\left(R\right)\leqq \left(m+1\right)$. Theorem 2: Suppose that $l.inj.{dim}_{R}P\leqq l.inj.{dim}_{R}R∕{N}^{2}<\infty$ for every projective $\left(R∕{N}^{2}\right)$-module $P$ and that $l.inj.{dim}_{R}{N}^{i-1}\ast {N}^{i}<\infty$ for $i\geqq 3$. Then $lFID\left(R\right)=l.inj.{dim}_{R}R<\infty$. 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 ${N}^{k-1}M\ne 0$ and ${N}^{k}M=0$, then every simple direct summand of ${N}^{k-1}$ is isomorphic to a direct summand of ${N}^{k-1}∕{N}^{k}$.

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