# Ring of infinite global dimension which does not have a finitely generated module of infinite projective dimension

Let $R$ be a ring of infinite global dimension. A priori we can't immediately conclude that $R$ has a module of infinite projective dimension, since it could be the case that $R$ only has a sequence of modules $(M_n)$ with $\text{p.dim} \hspace{2pt} M_n \geqslant n$. However, we then have $\text{p.dim} \hspace{2pt} (\bigoplus_n M_n) = \infty$, so there in fact does exist of module of infinite projective dimension. However, such a module need not be finitely generated.

Is there a ring $R$ of infinite global dimension which does not have a finitely generated module of infinite projective dimension?

I wouldn't be surprised if there was an obvious candidate for $R$ that I've missed.

• Edited. Thanks, Dec 31 '15 at 10:21

A ring $R$ for which every finitely generated right $R$-module has finite projective dimension is sometimes called right regular. Such rings are discussed in McConnell, Robson, Noncommutative Noetherian Rings, Chapter 7.7. In particular, Example 7.7.2 (6.4.9) gives an example of a ring of infinite global dimension which is left and right regular. The ring is constructed as a localization of a polynomial ring in countably many indeterminates. Maybe it is interesting to add that the ring is even Noetherian.