Let $X=Spec(R)$ be an irreducible noetherian scheme and $\eta$ the unique minimal prime ideal of $R$. Let $U$ and $V$ be open sets in $X$ and $p$ a point with $p\in V\subseteq U\subseteq X$.
If $X$ is additionally integer (i.e. additionally reuced, i.e. $R$ is a domain and $\eta=(0)$) then one has a diagram
$ \begin{array}{ccccccl} \mathcal{O}_X(U)& \hookrightarrow &\mathcal{O}_X(V)& \hookrightarrow&{\mathcal{O}}_{X,p}&\twoheadrightarrow& k(p)=Frac(R/p)&\\ &&&\searrow&\downarrow&&&\\ &&&&{\mathcal{O}}_{X,\eta}&\xrightarrow{=}& k(\eta)=Frac(R/\eta)&(=Frac(R)) \end{array} $
where the indicated maps ($\hookrightarrow$), the diagonal map ($\searrow$) and ${\mathcal{O}}_{X,p}\hookrightarrow {\mathcal{O}}_{X,\eta}$ are inclusions. I have some questions relating to this situation.
$\mbox{1.}$ Is the map $k(p)\to k(\eta)$ an inclusion?
One may identify all the images of the inclusions in ${\mathcal{O}}_{X,\eta}=Frac(R)$ with their domains and has $\mathcal{O}_X(U)=\bigcap_{p\in U}{\mathcal{O}}_{X,p}.~~(*)$
This identification is very helpful for me since one can really "work" then inside the big ring ${\mathcal{O}}_{X,\eta}$.
$\mbox{2.}$ I would like to understand function fields and stalks in the non-reduced case (but $X$ still irreducible). Then one can write down the same diagram as above (instead of the equality $Frac(R/\eta)=Frac(R)$). Which of the arrows remain inclusions ? Can I write down something like $(*)$ in this case, too?