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...
7 KB (1,222 words) - 09:42, 16 April 2025
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...
19 KB (2,748 words) - 00:15, 5 January 2025
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...
6 KB (775 words) - 00:21, 19 May 2025
("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...
7 KB (1,235 words) - 16:13, 27 May 2025
the term minimal prime may refer to Minimal prime ideal, in commutative algebra Minimal prime (recreational mathematics), the minimal prime number satisfying...
221 bytes (58 words) - 23:03, 21 November 2022
Krull dimension (redirect from Height of a prime ideal)
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...
11 KB (1,736 words) - 23:00, 7 May 2025
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...
20 KB (3,126 words) - 13:41, 17 April 2025
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...
10 KB (1,633 words) - 05:27, 20 May 2025
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...
26 KB (4,366 words) - 02:50, 26 March 2025
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...
16 KB (2,528 words) - 00:36, 7 April 2025
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...
3 KB (421 words) - 23:30, 26 November 2024
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...
6 KB (817 words) - 06:53, 11 July 2024
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...
3 KB (452 words) - 09:05, 7 November 2023
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...
10 KB (1,451 words) - 07:22, 28 May 2025
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...
12 KB (2,131 words) - 09:53, 19 November 2024
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...
38 KB (6,198 words) - 10:42, 15 May 2025
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...
6 KB (886 words) - 16:20, 29 January 2024
Glossary of commutative algebra (redirect from Saturated ideal)
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...
66 KB (9,772 words) - 22:19, 27 May 2025
Commutative ring (section Prime ideals)
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...
41 KB (5,688 words) - 11:41, 25 May 2025
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...
6 KB (979 words) - 20:48, 5 March 2025
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...
82 KB (12,496 words) - 00:02, 12 April 2025
Tight closure (category Ideals (ring theory))
{\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...
3 KB (469 words) - 22:54, 12 August 2023
points with maximal closure, that is the minimal prime ideals. The closed points correspond to maximal ideals of A. However, the spectrum and projective...
18 KB (2,787 words) - 03:12, 28 May 2025
Spectrum of a ring (redirect from Prime ideal topology)
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...
25 KB (4,089 words) - 21:05, 8 March 2025
≤ 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...
5 KB (957 words) - 22:20, 25 March 2025
Nilradical of a ring (category Ideals (ring theory))
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...
7 KB (1,066 words) - 10:27, 20 February 2025
their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural...
117 KB (14,179 words) - 16:20, 4 May 2025
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...
7 KB (951 words) - 00:21, 17 March 2025
respect to the field trace. It then encodes the ramification data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882...
8 KB (1,030 words) - 13:25, 31 March 2025
prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals...
3 KB (398 words) - 14:52, 10 February 2024