 Algebraically prime extension

For a first-order theory $T$ and $A\models T_{\forall }$, an algebraically prime extension is a structure embedding $i:A\to P\models T$ such that every embedding $j:A\to N\models T$ factors as $j=h\circ i$ for some structure embedding $h:P\to N$. The embeddings need not be elementary or unique. A theory has algebraically prime models when every such $A$ has an extension of this kind.