prime ideal

Noun

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Noun

**prime ideal** (*plural* prime ideals)

- (
*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, 1975, Springer, 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).

- Given a prime number p, there is only a finite number of
**1970**[Frederick Ungar Publishing], John R. Schulenberger (translator), Bartel Leendert 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}\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}.

- In the rings studied in Section 17.4 a nonzero
**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.

- In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the

- 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.

- Italian: ideale primo

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