scholarly journals On symbolic powers of prime ideals

2003 ◽  
pp. 329-342
Author(s):  
Sean Sather-Wagstaff
Keyword(s):  
Prime Ideals ◽  
Symbolic Powers

scholarly journals Symbolic powers of prime ideals with applications to hypersurface rings

1989 ◽  
Vol 113 ◽  
pp. 161-172 ◽  
Author(s):  
Sam Huckaba
Keyword(s):  
Prime Ideal ◽  
Complete Intersection ◽  
Fundamental Problem ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Commutative Noetherian Ring ◽  
Symbolic Powers ◽  
Associated Prime

Let R be a commutative Noetherian ring and suppose q is a prime ideal of R. A fundamental problem is to decide when powers qn of q are primary (that is qn is its own primary decomposition). If q is generated by a regular sequence then powers of q are always primary, because G(q, R) (the associated graded ring of R with respect to q) is an integral domain (see [12 page 98] and also [5 (2.1)]). Let qn) denote the nth symbolic power of q-defined by q(n) = {rεR|there exists sεR\q such that sr ε qn}. Then qn is primary if and only if qn = q(n) If q is generated by a regular sequence then we call it a complete intersection prime ideal, so if q is a complete intersection prime ideal then qn ≠ q(n) for all n ≥ 1. If q is not a complete intersection then powers need not be primary. If R is a three-dimensional regular local ring and q is a non-complete intersection height two prime ideal for example, then Huneke showed [11 Corollary (2.5)] that qn = q(n) for all n ≥ 2. Thus, for such a prime q it is impossible for qn to occur in the primary decomposition of any ideal. This phenomena increases the difficulty in finding a primary decomposition for an ideal having q as an associated prime.

scholarly journals On ordinary and symbolic powers of prime ideals

1986 ◽  
Vol 103 (1) ◽  
pp. 256-266
Author(s):  
Orlando E. Villamayor (H)
Keyword(s):  
Prime Ideals ◽  
Symbolic Powers

scholarly journals A Zariski--Nagata theorem for smooth ℤ-algebras

2020 ◽  
Vol 2020 (761) ◽  
pp. 123-140 ◽  
Author(s):  
Alessandro De Stefani ◽  
Eloísa Grifo ◽  
Jack Jeffries
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Classical Result ◽  
Differential Operators ◽  
Symbolic Power ◽  
Prime Ideals ◽  
Perfect Field ◽  
Mixed Characteristic ◽  
Symbolic Powers

AbstractIn a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra.

scholarly journals Links of symbolic powers of prime ideals

2007 ◽  
Vol 256 (4) ◽  
pp. 749-756 ◽  
Author(s):  
Hsin-Ju Wang
Keyword(s):  
Prime Ideals ◽  
Symbolic Powers

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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