Let D be a divison ring, let V be a left vector space of over D, possibly infinite dimensional, and let F be the prime field of D.
Is it true that every F-hyperplane of V contains a D-hyperplane of V?
I am mostly interested in the case when D has prime characteristic.
The answer is positive if D is finite, regardless of the dimension of V, although it is easier to see if V finite dimensional. Any help with with the case when D is infinite of prime characteristic would be appreciated.
The following should hopefully be an easy counterexample.
Let $F$ be a field, $D$ its purely trascendental extension $F(x)$ (field of fractions of $F[x]$, the $F$-polynomials in $x$), $V$ the field $F((x))$ of formal Laurant series (field of fractions of $F[[x]]$, formal power series in $x$). The direct sum decomposition $F((x))=F[x^{-1}]x^{-1}\oplus F\oplus xF[[x]]$ gives the $F$-hyperplane of formal Laurant series with 0 constant coefficient. Does it contain a $D$-hyperplane?
This would be impossible if every nonzero Laurant series $s((x))$ with $s((0))=0$ produces another $s'((x))$ with $s'((0))\neq0$ by multiplication and division with polynomials in $x$. Well, in fact it is enough multiplication or division by a power of $x$ (the power corresponding to a nonzero coefficient of $s((x))$).
