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I read that the global dimension of Z/4Z is not finite, I think that It's because that 4=2*2 and (2,2)!=1 hence Z/2Z direct sum with Z/2Z is not Z/4Z. Is it the reason for this ? If it's not to complicated, I would really like to see an explanation for why the global dimension is not finite. |
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It's not too complicated (I got it from [Lam: Lectures on modules and rings, Lemma 5.16], who claims it is an idea of Kaplansky.): Consider the exact sequence $0\to Z/2Z\stackrel{\cdot 2}{\to} Z/4Z\to Z/2Z\to 0$. Then $Z/2Z$ is not projective, since it is not a direct summand of $Z/4Z$ as you remarked. Therefore this yields that $Z/2Z$ has infinite projective dimension. |
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