prime ideal

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prime ideal

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

prime ideal (plural prime ideals)

1. (algebra, ring theory) 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 ${\displaystyle Q\subseteq I}$.
   • 1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39:

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

   • 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 ${\displaystyle {\mathfrak {o}}}$ on the basis of Axiom II; thus, in that section there are no lower prime ideals but ${\displaystyle {\mathfrak {o}}}$. Since every ideal ${\displaystyle {\mathfrak {a}}\neq {\mathfrak {o}}}$ is divisible by a prime ideal distinct from ${\displaystyle {\mathfrak {o}}}$ (proof: from among all the divisors of a distinct from ${\displaystyle {\mathfrak {o}}}$ choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to ${\displaystyle {\mathfrak {o}}}$.

   • 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 ${\displaystyle P}$ in a commutative ring ${\displaystyle R}$ is prime if, whenever we have two elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle R}$ such that ${\displaystyle ab\in P}$, it follows that ${\displaystyle a\in P}$ or ${\displaystyle b\in P}$; equivalently, ${\displaystyle P}$ is a prime ideal if and only if the factor ring ${\displaystyle R/P}$ is a domain.

2. 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 ${\displaystyle b\in I}$.