Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}\_{\ge 0}$ with respect to which $F$ is complete. Let $\mathcal{O}\_{F}$ denote the ring of integers of $F$.



> 
> First define the product norm $|\cdot|\_{prod}$ on $F\otimes \_{\mathbb F\_p} F$ in the following way. If $c\in F\otimes \_{\mathbb F\_p} F$, then
> 
> 
> $$|c|\_{prod}:=\inf\left\{\max\_{1\le i\le n}\{|c\_{1,i}||c\_{2,i}| \}\ : \ c=\sum^{n}\_{i=1}c\_{1,i}\otimes c\_{2,i}, \text{where } c\_{1,i}, c\_{2,i}\in F\right\}$$
> On the other hand, define $|\cdot|'\_{prod}$ on the subring $\mathcal{O}\_{F}\otimes \_{\mathbb F\_p} 
> \mathcal{O}\_{F}$ in the following way. If $d\in \mathcal{O}\_{F}\otimes \_{\mathbb F\_p} 
> \mathcal{O}\_{F}$, then
> 
> 
> $$|d|'\_{prod}:=\inf\left\{\max\_{1\le i\le n}\{|d\_{1,i}||d\_{2,i}| \}\ : \ d=\sum^{n}\_{i=1}d\_{1,i}\otimes d\_{2,i}, \text{where } d\_{1,i}, d\_{2,i}\in \color{red}{\mathcal{O}\_{F}}\right\}$$
> 
> 
> 



> 
> In both definitions the infimum is taken over all the possible ways to write the element as a sum of pure tensors in the respective rings.
> Is it true that $|\cdot|\_{prod}$ and $|\cdot|'\_{prod}$ coincide when restricting them to elements of $\mathcal{O}\_{F}\otimes \_{\mathbb F\_p} 
> \mathcal{O}\_{F}$?
> 
> 
> 


My first guess is that the answer is true. It is clear that $|c|\_{prod}\le |c|'\_{prod}$ for $c\in \mathcal{O}\_{F}\otimes\_{\mathbb{F}\_{p}}\mathcal{O}\_{F}$, since there are more ways to write $c$ as a sum of pure tensors in $F$.


I am able to show that they do coincide on pure tensors in $\mathcal{O}\_{F}\otimes \_{\mathbb F\_p} 
\mathcal{O}\_{F}$, more precisely I showed that $|x\otimes y|\_{prod}=|x\otimes y|'\_{prod}=|x|\cdot|y|$ for $x,y\in \mathcal{O}\_{F}$, but I am not able to prove that in the general case.



Product absolute value in rings of integers
The natural map $i \colon \mathcal O\_F \to F$ is an injective map of $\mathbb F\_p$-vector spaces, hence we can choose a splitting, i.e. an $\mathbb F\_p$-linear map $s \colon F \to \mathcal O\_F$ such that $s \circ i$ is the identity.


We have $|s(x)| \leq |x|$ since if $x \in \mathcal O\_F$ then $|s(x) | = |x|$ and if $x \notin \mathcal O\_F$ then $|s(x)| \leq 1 \leq |x|$.


By $\mathbb F\_p$-linearity, if $$c = \sum\_{i=1}^n c\_{1,i} \otimes c\_{2,i}$$ then $$ s \otimes s(c) = \sum\_{i=1}^n s(c\_{1,i}) \otimes s(c\_{2,i})$$ and if $c \in \mathcal O\_F \otimes \mathcal O\_F$ then by definition $$s \otimes s(c)=c.$$


Combining these, we have


$$\begin{aligned}|c|\_{prod}
&=\inf\left\{\max\_{1\le i\le n}\{|c\_{1,i}||c\_{2,i}| \}\ : \ c=\sum^{n}\_{i=1}c\_{1,i}\otimes c\_{2,i}, \text{where } c\_{1,i}, c\_{2,i}\in F\right\}
\\
&\ge \inf\left\{\max\_{1\le i\le n}\{|s(c\_{1,i})||s(c\_{2,i})| \}\ : \ c=\sum^{n}\_{i=1}c\_{1,i}\otimes c\_{2,i}, \text{where } c\_{1,i}, c\_{2,i}\in F\right\}
\\
 &= \inf\left\{\max\_{1\le i\le n}\{|s(c\_{1,i})||s(c\_{2,i})| \}\ : \ s\otimes s (c)=\sum^{n}\_{i=1}s(c\_{1,i})\otimes s(c\_{2,i}), \text{where } c\_{1,i}, c\_{2,i}\in F\right\} 
\\
&= \inf\left\{\max\_{1\le i\le n}\{|s(c\_{1,i})||s(c\_{2,i})| \}\ : \ c=\sum^{n}\_{i=1}s(c\_{1,i})\otimes s(c\_{2,i}), \text{where } c\_{1,i}, c\_{2,i}\in F\right\} 
\\
&\ge \inf\left\{\max\_{1\le i\le n}\{|d\_{1,i}||d\_{2,i}| \}\ : \ c=\sum^{n}\_{i=1}d\_{1,i}\otimes d\_{2,i}, \text{where } d\_{1,i}, d\_{2,i}\in {\mathcal{O}\_{F}}\right\} 
\\
&= |c|'\_{prod}\end{aligned}$$


which together with the trivial inequality proves the identity.


