• 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
  • Thumbnail for Prime ideal
    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
  • 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} 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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...
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  • Thumbnail for Zariski topology
    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
  • 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
  • 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...
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  • 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
  • Thumbnail for Prime number
    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