prime ideal

Etymology
By analogy with the notion of in number theory.

Noun

 * 1)  Any (two-sided) ideal $$I$$ such that for arbitrary ideals $$P$$ and $$Q$$, $$PQ\subseteq I\implies P\subseteq I$$ or $$Q\subseteq I$$.
 * 2) * 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator),, Algebra, Volume 2, 2003, Springer, page 189,
 * In the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by $$\mathfrak{o}$$ on the basis of Axiom II; thus, in that section there are no lower prime ideals but $$\mathfrak{o}$$. Since every ideal $$\mathfrak{a}\ne\mathfrak{o}$$ is divisible by a prime ideal distinct from $$\mathfrak{o}$$ (proof: from among all the divisors of a distinct from $$\mathfrak{o}$$ choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to $$\mathfrak{o}$$.
 * 1) In a commutative ring, a (two-sided) ideal $$I$$ such that for arbitrary ring elements $$a$$ and $$b$$, $$ab \in I \implies a \in I $$ or $$b \in I$$.
 * 1) In a commutative ring, a (two-sided) ideal $$I$$ such that for arbitrary ring elements $$a$$ and $$b$$, $$ab \in I \implies a \in I $$ or $$b \in I$$.
 * 1) In a commutative ring, a (two-sided) ideal $$I$$ such that for arbitrary ring elements $$a$$ and $$b$$, $$ab \in I \implies a \in I $$ or $$b \in I$$.

Translations

 * Italian: ideale primo