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Keywords:
Wakamatsu tilting module; $\omega $-$k$-torsionfree module; $\mathcal {X}$-resolution dimension; injective dimension; $\omega $-torsionless property
Wakamatsu tilting module; $\omega $-$k$-torsionfree module; $\mathcal {X}$-resolution dimension; injective dimension; $\omega $-torsionless property
Summary:
Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_R\omega $ a Wakamatsu tilting module with $S={\rm End}(_R\omega )$. We introduce the notion of the $\omega $-torsionfree dimension of finitely generated $R$-modules and give some criteria for computing it. For any $n\geq 0$, we prove that ${\rm l.id}_R(\omega ) = {\rm r.id}_S(\omega )\leq n$ if and only if every finitely generated left $R$-module and every finitely generated right $S$-module have $\omega $-torsionfree dimension at most $n$, if and only if every finitely generated left $R$-module (or right $S$-module) has generalized Gorenstein dimension at most $n$. Then some examples and applications are given.
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