Question
If $R=\mathbb{Z}[x] /\left(x^{k}-1\right)$, prove that $\mathrm{D}(R)=\infty$ (where $\mathrm{D}$ is global dimension).
Step 1
In other words, it is the largest possible length of a projective resolution of any R-module. If there is no bound on the lengths of projective resolutions, then D(R) = ∞. Now, consider the ring R = ℤ[x] / (x^k - 1). We want to show that D(R) = ∞. To do this, we Show more…
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prove the following
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