domain and P is a nonzero fact about::prime ideal in R. Then, there exists ... * Prime ideal need not contain a prime element
===Applications=== ... ...
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unique factorization domain. Then, every prime ideal in R is generated by finitely ... * Unique factorization implies every nonzero prime ideal contains ... ...
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domain R and a nonzero fact about::prime ideal P of R such that P does ... * Unique factorization implies every nonzero prime ideal contains ... ...
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* Weaker than::Minimal prime ideal * Weaker than::Principal prime ideal
===Weaker properties=== ... ...
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#redirect Codimension of a prime ideal ... ...
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Let A be a commutative unital ring and P a prime ideal in A. Then, the localization of A at P is defined as follows:
* As a set, it is the collection ... ...
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An ideal in a commutative unital ring is termed a minimal prime ideal ... * It is a prime ideal, and there is no prime ideal strictly contained ... ...
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An ideal in a commutative unital ring is termed a principal prime ... * It is a principal ideal and a prime ideal
* It is generated by a ... ...
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#redirect Codimension of a prime ideal ... ...
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Let R be a commutative unital ring, Q a prime ideal of R and n a positive integer. The n^{th} symbolic power of Q, denoted as Q^{(n)} is defined ... ...
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* Every fact about::prime ideal in the ring is a principal ideal. a commutative unital ring in which every prime ideal is principal, and a principal ... ...
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domain) every fact about::prime ideal is generated by finitely ... * Unique factorization and Noetherian implies every prime ideal is ... ...
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unital ring satisfying the following: every prime ideal of the ring contains a minimal prime ideal.
==Relation with other properties== ... ...
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* Every fact about::prime ideal contains a fact about::minimal prime ideal ... ===Prime ideal===
Prime ideal
===Minimal prime ideal=== ... ...
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Here, M_P denotes the localization of M at the prime ideal P. ... * If a prime ideal P is contained in the support of M, then any prime ... ...
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* Every fact about::prime ideal in the ring is a principal ideal. a commutative unital ring in which every prime ideal is principal, and a principal ... ...
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radical if its radical is a defining ingredient::prime ideal. ... | Weaker than::prime ideal || product of two elements in the ideal ... ...
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==Statement==
Suppose (A,\mathfrak{m}) is a Noetherian local ring that is not Artinian: in other words, the unique maximal ideal \mathfrak{m} ... ...
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that is not contained in any fact about::minimal prime ideal. ... by x: Since x is nilpotent, any prime ideal must contain x, and hence ... ...
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* If I is a prime ideal, it is defined as the Krull dimension of the localization R_I
* Otherwise, it is defined to be the minimum of the Krull ... ...
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and if x \notin N, then there exists a prime ideal not containing x. Consider ... * Any such ideal must be a prime ideal
Thus, we have found a prime ... ...
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An ideal in a commutative unital ring is termed a principal prime ... * It is a principal ideal and a prime ideal
* It is generated by a ... ...
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An ideal in a commutative unital ring is termed a minimal prime ideal ... * It is a prime ideal, and there is no prime ideal strictly contained ... ...
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Any maximal ideal in a commutative unital ring is a prime ideal. ... ===Prime ideal===
Prime ideal
==Proof==
===As quotient-determined properties ... ...
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must be inside some fact about::minimal prime ideal of A. every zero divisor must be inside some prime ideal, since minimal primes ... ...
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* Every prime ideal in it is either a minimal prime ideal or a maximal ... ring where there is a unique minimal prime ideal: the zero ideal ... ...
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* Prime ideal in a unique factorization domain. Unique factorization implies every prime ideal is generated by prime ... ...
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a one-dimensional domain: every nonzero prime ideal in it is maximal. ... # uses::Principal ideal ring iff every prime ideal is principal ... ...
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* Every prime ideal of the ring contains a minimal prime ideal
* There exist only finitely many minimal prime ideals
Equivalently, it is a ring ... ...
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Let R be a commutative unital ring, Q a prime ideal of R and n a positive integer. The n^{th} symbolic power of Q, denoted as Q^{(n)} is defined ... ...
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# If P is a prime ideal of R and if S := R/P contains an element b ... As in the hypotheses, let P be a prime ideal in R. Then S := R/P is ... ...
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==Definition==
Let R be a commutative unital ring. A collection of prime ideals in R is termed a complete system of prime ideals for R if R is ... ...
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