Abstract
Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n \geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_{R}^{i}(M, R)=0$ for any $1 \leq i \leq n$. Then we investigate the relation between these properties and the self-injective dimension of $R$.
Citation
Chonghui Huang. Zhaoyong Huang. "Torsionfree dimension of modules and self-injective dimension of rings." Osaka J. Math. 49 (1) 21 - 35, March 2012.
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