algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers...

In mathematics, the prime ideal theorem may be the Boolean prime ideal theorem the Landau prime ideal theorem on number fields This disambiguation page...

more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese...

In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement...

after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More...

commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R {\displaystyle R} is the set of all prime ideals of R {\displaystyle...

ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal...

primary ideal is a prime ideal. This concept is generalized to non-commutative rings in the semiprime ring article. The radical of an ideal I {\displaystyle...

theorem use minimal prime ideals. A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I....

a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest...

their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural...

maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain....

all prime ideals and the second over the maximal ideals. There is a bijection between the set of prime ideals of S−1R and the set of prime ideals of R...

the ideal (2, 1 + √-5) is a prime ideal which cannot be generated by a single element. Historically, the idea of factoring ideals into prime ideals was...

("Principal") + ideal + Satz ("theorem")). Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal containing...

(pn) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian...

primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by Emanuel Lasker (1905) for the special...

a nonzero prime ideal. (Note that in an integral domain, the ideal (0) is a prime ideal, but 0 is an exception in the definition of 'prime element'.)...

number field) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals. Any maximal ideal is a prime ideal or, more briefly...

in algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let O K {\displaystyle...

In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula...

principal ideal domain is Noetherian. In all unital rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal...

the set of a ∈ K with v(a) ≥ 0, the prime ideal mv is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of Rv), the residue field kv = Rv/mv...

algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set...

example, if R is the localization of k[x, y, z]/(x2 + y3 + z7) at the prime ideal (x, y, z) then R is a local ring that is a UFD, but the formal power...

primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological...

divisors. An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal. An integral domain is a nonzero commutative ring for which every...

to a prime ideal in a polynomial ring, and the points of such an affine variety correspond to the maximal ideals that contain this prime ideal. The Zariski...

such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case...

it is prime (that is, it generates a prime ideal). The prime elements of Z[i] are also known as Gaussian primes. An associate of a Gaussian prime is also...