= Algebraically prime extension
{title2=$j=h\circ i$}
= Algebraically prime model
{synonym}
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.
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