What is the meaning of Prime_ideal?

Any (two-sided) ideal ${\displaystyle I}$ such that for arbitrary ideals ${\displaystyle P}$ and ${\displaystyle Q}$, ${\displaystyle PQ\subseteq I\implies P\subseteq I}$ or ${\displaystyle Q\subseteq I}$.

In a commutative ring, a (two-sided) ideal ${\displaystyle I}$ such that for arbitrary ring elements ${\displaystyle a}$ and ${\displaystyle b}$, ${\displaystyle ab\in I\implies a\in I}$ or ${\displaystyle b\in I}$.

Source: wiktionary.org