# tensorproduct, p-adic groupring

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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For every ring homomorphism $R \to S$ and for every monoid $G$ we have $S[G] \cong S \otimes_R R[G]$. Your question is not research level and therefore will be probably closed. See also the FAQ. –  Martin Brandenburg Nov 17 '12 at 14:37
This is not only far from a research question, it's even below master level. It should already be closed! –  Fernando Muro Nov 17 '12 at 14:55

## 1 Answer

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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