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Keywords:
global dimension; $\ast $-module
global dimension; $\ast $-module
Summary:
In this note we show that for a $\ast ^{n}$-module, in particular, an almost $n$-tilting module, $P$ over a ring $R$ with $A=\mathop {\mathrm End}_{R}P$ such that $P_A$ has finite flat dimension, the upper bound of the global dimension of $A$ can be estimated by the global dimension of $R$ and hence generalize the corresponding results in tilting theory and the ones in the theory of $\ast $-modules. As an application, we show that for a finitely generated projective module over a VN regular ring $R$, the global dimension of its endomorphism ring is not more than the global dimension of $R$.
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