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
    $\begingroup$ The title of this good question is a good example of the ambiguity of “any” in mathematical English. I think it would be better to use “every” — I first read the title as asking if some graded hereditary ring is hereditary. $\endgroup$ Oct 3 at 12:45
  • $\begingroup$ @Carl-Fredrik Nyberg Brodda Thank you for this comment. I have changed the title. $\endgroup$
    – Ralle
    Oct 3 at 14:56

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.