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...

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...

the poset of prime ideals of a ring, which may include the zero ideal as a minimal prime ideal. The definition of a minimal right ideal N of a ring R is...

("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...

catenary. In a Noetherian ring, a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by n elements (Krull's...

the term minimal prime may refer to Minimal prime ideal, in commutative algebra Minimal prime (recreational mathematics), the minimal prime number satisfying...

ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are...

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...

module theory). The minimal elements of { Q i ∣ i } {\displaystyle \{{\sqrt {Q_{i}}}\mid i\}} are the same as the minimal prime ideals containing I {\displaystyle...

Maximal right/left/two-sided ideals are the dual notion to that of minimal ideals. If F is a field, then the only maximal ideal is {0}. In the ring Z of integers...

A} such that for each prime ideal p, the completion of the localization Ap is equidimensional, i.e. for each minimal prime ideal q in the completion A...

of prime ideals p , q {\displaystyle {\mathfrak {p}},{\mathfrak {q}}} in it, for each prime ideal r {\displaystyle {\mathfrak {r}}} that is a minimal prime...

is a direct product of r Krull domains, where r is the number of minimal prime ideals of A. The theorem is a partial generalization of the Krull–Akizuki...

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...

all zero-divisors in a reduced ring R. Then D is the union of all minimal prime ideals. Over a Noetherian ring R, we say a finitely generated module M has...

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...

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...

would then contradict the minimality requirement on f, so f must be irreducible. The minimal polynomial f of α generates the ideal Jα, i.e. every g in Jα...

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 minimal prime ideals p i Q ( A ) {\displaystyle...

prime ideals and points in an affine scheme, this means X is irreducible iff X is connected and the rings Ai all have exactly one minimal prime ideal...

it is the set of all prime ideals that contain some prime ideal, and the irreducible components correspond to minimal prime ideals. The number of irreducible...

of R is irreducible. R possesses exactly one minimal prime ideal (this prime ideal may be the zero ideal); If R is subdirectly irreducible or meet-irreducible...

as the intersection of all the prime ideals of the ring (in fact, it is the intersection of all minimal prime ideals). Proposition— Let R {\displaystyle...

points with maximal closure, that is the minimal prime ideals. The closed points correspond to maximal ideals of A. However, the spectrum and projective...

of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimal prime ideals (i.e., the ones...

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...

{\displaystyle c\in R} , where c {\displaystyle c} is not contained in any minimal prime ideal of R {\displaystyle R} , such that c z p e ∈ I [ p e ] {\displaystyle...

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

with kernel a prime ideal KP. These prime ideals are in bijection with the orderings Xk of k and constitute the minimal prime ideal spectrum MinSpec W(k)...

same as that of R. minimal 1.  A minimal prime of an ideal is a minimal element of the set of prime ideals containing it. 2.  A minimal resolution of a module...