prime ideal

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 ${\\displaystyle I}$ 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, 1975, Springer, 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 ${\\displaystyle {\\mathfrak {a}}\eq {\\mathfrak {o}}}$ 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 ${\\displaystyle b\\in I}$.

Translations

(ring theory) type of ideal

Italian: ideale primo m