In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal prime ideals.
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the poset of prime ideals of a ring, which may include the zero ideal as a minimalprimeideal. The definition of a minimal right ideal N of a ring R is...
algebra, a primeideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The primeideals for the integers...
the term minimalprime may refer to Minimalprimeideal, in commutative algebra Minimalprime (recreational mathematics), the minimalprime number satisfying...
catenary. In a Noetherian ring, a primeideal has height at most n if and only if it is a minimalprimeideal over an ideal generated by n elements (Krull's...
ring have only one minimalprime. It follows that the unique minimalprimeideal of a reduced and irreducible ring is the zero ideal, so such rings are...
module theory). The minimal elements of { Q i ∣ i } {\displaystyle \{{\sqrt {Q_{i}}}\mid i\}} are the same as the minimalprimeideals containing I {\displaystyle...
Maximal right/left/two-sided ideals are the dual notion to that of minimalideals. If F is a field, then the only maximal ideal is {0}. In the ring Z of integers...
primary ideal is a primeideal. This concept is generalized to non-commutative rings in the semiprime ring article. The radical of an ideal I {\displaystyle...
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all primeideals of R, and is usually denoted...
algebra, an associated prime of a module M over a ring R is a type of primeideal of R that arises as an annihilator of a (prime) submodule of M. The set...
all zero-divisors in a reduced ring R. Then D is the union of all minimalprimeideals. Over a Noetherian ring R, we say a finitely generated module M has...
of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimalprimeideals (i.e., the ones...
their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and primeideals. A natural...
Thus, any proper ideal I of Q(A) is contained in the set of zero divisors of Q(A); that set equals the union of the minimalprimeideals p i Q ( A ) {\displaystyle...
points with maximal closure, that is the minimalprimeideals. The closed points correspond to maximal ideals of A. However, the spectrum and projective...
as the intersection of all the primeideals of the ring (in fact, it is the intersection of all minimalprimeideals). Proposition — Let R {\displaystyle...
ring, there are only finitely many minimalprimeideals. Also, the descending chain condition holds on primeideals. In a commutative Noetherian domain...
it is the set of all primeideals that contain some primeideal, and the irreducible components correspond to minimalprimeideals. The number of irreducible...
it is prime (that is, it generates a primeideal). The prime elements of Z[i] are also known as Gaussian primes. An associate of a Gaussian prime is also...
universal. Examples of valency frames for the "say" semantic prime: someone said something→[minimal frame] someone said: '––'→[direct speech] someone said something...
same as that of R. minimal 1. A minimalprime of an ideal is a minimal element of the set of primeideals containing it. 2. A minimal resolution of a module...
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