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Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite?

Similarly, Let $R$ be ring such that every left $R$-module has finite flat dimension. Is the weak global dimension of $R$ finite?

user26857
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Aimin Xu
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1 Answers1

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Yes. If $R$ has infinite left global dimension, then for every $n$ there is a left module $M_n$ with projective dimension greater than $n$. But then $\bigoplus_{n\in\mathbb{N}}M_n$ has infinite projective dimension.

The same argument works for weak dimension.

Jeremy Rickard
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