Suppose there is a cyclic group $G$ and a prime $p$. Why can one write
$$ \mathbb{Z}_p[G] \cong \mathbb{Z}_p \otimes _\mathbb{Z} \mathbb{Z}[G]$$
Is this some theorem which has a name? Thanks for hints.
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I do not think this is a research question. Anyway, first of all you should convince yourself that $\mathbb Z_p\otimes_{\mathbb Z} (\mathbb Z[G])\cong(\mathbb Z_p\otimes_{\mathbb Z} \mathbb Z)[G]$. After this I think you will have no difficulty in verifying that $\mathbb Z_p\otimes_{\mathbb Z} \mathbb Z\cong \mathbb Z_p$. This is more an exercise than a theorem so it has no specific name. Anyway, similar operations are usually called "extensions of scalars". |
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