$\DeclareMathOperator\Spec{Spec}\newcommand\Ring{\mathrm{Ring}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}$In the category of schemes the objects of the form $\Spec(K)$ with $K$ a field can be characterized as follows. They are precisely the non-empty schemes which have no proper subobjects in $Sch$.

Consider now the category $\Ring\op$ with its various Grothendieck topologies (Zariski, étale, fpqc, etc.). From those we get various big sheaf topoi which contain the category of schemes as a full subcategory. As a functor, $\Spec(K)$ is the representable presheaf $\Hom_{\Ring}(K,-)$.

Even though $\Spec(K)$ has no proper subschemes when $K$ is a field, it can have non-trivial sub-pre-sheaves. Can it have non-trivial Zariski sub-sheaves? If yes, can we switch to one of the finer topologies to prevent this? Is one of the topologies on $\Ring\op$ fine enough, so that the sheaves of the form $\Spec(K)$ with $K$ a field are precisely the objects with a trivial subobject lattice in $\Sh(\Ring\op,\text{sth})$?