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finitistic dimension; restricted injective dimension; tilting module
We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$-module with $A=\mathop {\rm End}_RT$. If $_RT$ is selforthogonal, then we show that $\mathop {\rm rid}(T_A)\leq \mathop {\rm findim}(A_A)\leq \mathop {\rm findim}(_RT)+\mathop {\rm rid}(T_A)$. Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left \mbox {$R$-module} with finite injective dimension, then $\mathop {\rm rid}(T_A)\leq \mathop {\rm findim}(A_A)\leq \mathop {\rm fin.inj.dim}(_RR)+\mathop {\rm rid}(T_A)$. Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.
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