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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

{\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...

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...

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...

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...

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 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...

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

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...

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...

=I^{m+k}} . J and I have the same radical and the same set of minimal prime ideals over them (the converse is false). If R is a Noetherian ring, then...

≤ r {\displaystyle {\mathfrak {p}}_{i},\,1\leq i\leq r} are the minimal prime ideals of A, then the total ring of fractions K of A is the direct product...