prime ideal

Meaning

Noun

• Any (two-sided) ideal $I$ such that for arbitrary ideals $P$ and $Q$, $PQ\backslash subseteq\; I\backslash implies\; P\backslash subseteq\; I$ or $Q\backslash subseteq\; I$.
• In a commutative ring, a (two-sided) ideal $I$ such that for arbitrary ring elements $a$ and $b$, $ab\; \backslash in\; I\; \backslash implies\; a\; \backslash in\; I$ or $b\; \backslash in\; I$.

Origin

• By analogy with the notion of prime number in number theory.