Let R be a graded (associative, unital) ring. If R is left graded hereditary (i.e. its left graded global dimension is 0 or 1), does it follow that R is left hereditary (i.e. its left global dimension is 0 or 1)?

## 1 Answer

No, consider the graded ring $R_{\ast}=\mathbb{Z}[x,x^{-1}]$ with $|x|=1$. Then the functor $M\mapsto M_0$ gives an equivalence from graded $R_{\ast}$-modules to abelian groups, so $R_{\ast}$ is hereditary in the graded sense. However, the corresponding ungraded ring is not hereditary: for example, the inhomogeneous ideal generated by $(2,x-1)$ is not a free module.

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