I know that if we assume zero sharp we obtain for a class of standard ordinals
$L\_\alpha\prec L\_\beta$ , $\alpha <\beta$ and obviously $L\_\alpha\in L\_\beta$.


My question is :
If we only assume the existence of a standard transitive model ,and from it we have a countable standard ordinal $\alpha$, $L\_\alpha\models ZFC$, being for example $L\_\alpha$ the minimal model,then there exist $N$ elementary end-extension $L\_\alpha\prec N$ and $L\_\alpha\in N$?, $N$ not necessarily externally well-founded.


Any help or reference would be appreciated.



Elementary extension $L_\alpha\prec N$ such that $L_\alpha \in N$
The answer is no, there can be no such elementary end-extension $N$ of the minimal model $L\_\alpha$ with $L\_\alpha\in N$. The reason is that $N$ would have to be $\omega$-standard, since it has the same $\omega$ as $L\_\alpha$, and thus $N$ would have the correct understanding of the theory ZFC, and it would think that "there is a transitive model of ZFC". By elementarity, $L\_\alpha$ would have to think that also, but it cannot since this would violate minimality. 


This argument does not assume that $N$ is well-founded, but only that $L\_\alpha$ is both an element and a transitive substructure of $N$. 


