Tensor products over $\mathbb{Z}$ I am doing some computation using spectra and I would need to compute the following two tensor products:
$\mathcal{O}_K\otimes_{\mathbb{Z}} \mathbb{Q}$ and $\mathcal{O}_K\otimes_{\mathbb{Z}}\mathbb{F}_p$, where $\mathcal{O}_K$ is the ring of integers of a finite extension $\mathbb{Q}\leq K$ and $\mathbb{F}_p$ is the field with $p$ elements. I know that $\mathcal{O}_K$ is finitely generated as a $\mathbb{Z}$-module, but it cannot always be generated by a single element. Can you give me any suggestions?
 A: All tensor products are over $\mathbb{Z}$ unless otherwise specified. 
$\mathcal{O}_K$ is a free $\mathbb{Z}$-module of rank $[K\colon \mathbb{Q}]$. In fact, there exists a basis of $\mathcal{O}_K$ over $\mathbb{Z}$ that is also a basis of $K$ over $\mathbb{Q}$. Hence the naturally defined map
$$\mathcal{O}_K \otimes \mathbb{Q} \ni \sum o_l \otimes r_l \mapsto \sum o_l \cdot r_l \in K$$ is an isomorphism. In other words, $$\mathcal{O}_K \otimes \mathbb{Q} = K$$  In fact  for extensions of number fields $K/L$ with $L$ not necessarily $\mathbb{Q}$ we will have 
$$O_K \otimes_{O_L}L = K$$ Note that $O_K$ is not necessarily free over $O_L$ but it is locally free so the things still work. 
$\bf{Added}$  Say $R$ is a ring, $S^{-1} R$ a ring of fractions of $R$, $V$ a $S^{-1} R$-module, $\mathcal{L}$ an $R$-submodule of $V$. Then we have the natural isomorphism $$ \mathcal{L} \otimes_R S^{-1}R \simeq S^{-1} R \cdot \mathcal{L}$$
( RHS - combinations of elements of $\mathcal{L}$ with coefficients in $S^{-1}R$). Knowing this, we have the general result: 
$B$ domain, $L$ its field of fractions, $L \subset K$ an algebraic extension, $A$ the integral closure of $B$ in $K$. Then we have $K = L \cdot A$ and so $  A\otimes_B L= K$.
The other tensor product: 
$$\mathcal{O}_K \otimes \mathbb{F}_p = \mathcal{O}_K \otimes  \mathbb{Z}/ p \mathbb{Z} = \mathcal{O}_K / p \mathcal{O}_K$$
Note that in $\mathcal{O}_K / p \mathcal{O}_K$ the denominator is not only a subgroup of $\mathcal{O}_K$ but is the ideal generated by $p$. Now we want to bring in some algebraic number theory on how the ideal $p\mathcal{O}_K$ is a product of powers of  different prime ideals of $\mathcal{O}_K$. 
