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| Let $R$ be a left noetherian ring, $S$ a right noetherian ring and
$_RU$ a generalized tilting module with $S={\rm End}(_RU)$. The
injective dimensions of $_RU$ and $U_S$ are identical provided both
of them are finite. Under the assumption that the injective
dimensions of $_RU$ and $U_S$ are finite, that the subcategory $\{
{\rm Ext}_S^n(N, U)|N$ is a finitely generated right $S$-module$\}$
is submodule closed is characterized equivalently. From which, a
negative answer to a question posed by Auslander in 1969 is given.
Finally, some partial answers to Wakamatsu Tilting Conjecture are
given. |
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| Keywords:generalized tilting modules, injective dimension, $U$-limit dimension, submodule closed, Wakamatsu Tilting Conjecture |
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