prime ideal

English

Etymology

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

Noun

prime ideal (plural prime ideals)

1. (algebra, ring theory) Any (two-sided) ideal $I$ such that for arbitrary ideals $P$ and $Q$, $PQ\subseteq I\implies P\subseteq I$ or $Q\subseteq I$.
   • 1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39:

     Given a prime number $p$, there is only a finite number of prime ideals ${\mathfrak {p}}$ in ${\mathfrak {o}}$ such that ${\mathfrak {p}}\cap J=p$ (they are the prime ideals of ${\mathfrak {o}}p$).

   • 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, 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 ${\mathfrak {a}}\neq {\mathfrak {o}}$ 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 ${\mathfrak {o}}$.

   • 2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:

     In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal $P$ in a commutative ring $R$ is prime if, whenever we have two elements $a$ and $b$ of $R$ such that $ab\in P$, it follows that $a\in P$ or $b\in P$; equivalently, $P$ is a prime ideal if and only if the factor ring $R/P$ is a domain.

2. 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

(ring theory) type of ideal

• Italian: ideale primo m