scholarly journals U–essential prime divisors and sequences over an ideal

1986 ◽  
Vol 103 ◽  
pp. 39-66 ◽  
Author(s):  
Daniel Katz ◽  
Louis J. Ratliff
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Asymptotic Theory ◽  
Minimal Prime Ideal ◽  
Standard Theory ◽  
Noetherian Rings ◽  
Associated Primes ◽  
Prime Divisors ◽  
Minimal Prime

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

Form rings and projective equivalence

1986 ◽  
Vol 99 (3) ◽  
pp. 447-456 ◽  
Author(s):  
Daniel Katz ◽  
L. J. Ratliff
Keyword(s):  
Local Ring ◽  
Prime Divisor ◽  
Noetherian Ring ◽  
Noetherian Rings ◽  
Local Rings ◽  
Prime Divisors ◽  
Research Problems ◽  
Form Ring ◽  
Divisor Of Zero ◽  
Minimal Prime

If I and J are ideals in a Noetherian ring R, then I and J are projectively equivalent in case (Ii)a = (Jj)a for some positive integers i, j (where Ka denotes the integral closure in R of the ideal K) and the form ring F(R, I) of R with respect to I is the graded ring R/I ⊕ I/I2 ⊕ I2/I3 ⊕ …. These two concepts have played an important role in many research problems in commutative algebra, so they have been deeply studied and many of their properties have been discovered. In a recent paper [13] they were combined to show that a semi-local ring R is unmixed if and only if for every ideal J in R there exists a projectively equivalent ideal J in R such that every prime divisor of zero in F(R, J) has the same depth. It seems to us that results similar to this are interesting and potentially quite useful, so in this paper we prove several additional such theorems. Namely, it is shown that all ideals in all local rings have a projectively equivalent ideal whose form ring is fairly nice. Also, a characterization similar to the just mentioned result in [13] is given for the class of local rings whose completions have no embedded prime divisors of zero, and several analogous new characterizations are given for locally unmixed Noetherian rings. In particular, it is shown that if I is an ideal in an unmixed local ring R such that height(I) = l(I) (where l(I) denotes the analytic spread of I), then there exists a projectively equivalent ideal J in R such that Ass (F(R, J)) has exactly m elements, all minimal, where m is the number of minimal prime divisors of I (so if I is open, then F(R, J) has exactly one prime divisor of zero and is a locally unmixed Noetherian ring).

Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings

1981 ◽  
Vol 33 (2) ◽  
pp. 325-346 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Additional Assumption ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Local Rings ◽  
Minimal Prime

If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP/J(RP) is simple Artinian: in short it has several properties of the commutative local rings.In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x1, x2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x1, x2, …, xj (j = 0, 1, …, n), are all distinct primes.

Associated Prime Ideals in Non-Noetherian Rings

1984 ◽  
Vol 36 (2) ◽  
pp. 344-360 ◽  
Author(s):  
Juana Iroz ◽  
David E. Rush
Keyword(s):  
Prime Ideal ◽  
Finite Subset ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Associated Primes ◽  
The Past ◽  
Associated Prime Ideals ◽  
Associated Prime

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

scholarly journals On the vanishing and the positivity of intersection multiplicities over local rings with small non complete intersection loci

1994 ◽  
Vol 136 ◽  
pp. 133-155 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Prime Ideal ◽  
Local Rings ◽  
Regular Local Ring ◽  
Commutative Noetherian Ring ◽  
Intersection Multiplicities ◽  
Minimal Prime

Throughout this paperAis a commutative Noetherian ring of dimensiondwith the maximal ideal m and we assume that there exists a regular local ringSsuch thatAis a homomorphic image ofS, i.e.,A=S/Ifor some idealIofS. Furthermore we assume thatAis equi-dimensional, i.e., dimA= dimA/for any minimal prime idealofA. We put.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

scholarly journals Primary Decomposition of Ideals of Lattice Homomorphisms

10.37236/7694 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Leila Sharifan ◽  
Ali Akbar Estaji ◽  
Ghazaleh Malekbala
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Minimal Prime Ideal ◽  
Primary Decomposition ◽  
Planar Lattice ◽  
Combinatorial Properties ◽  
Finite Lattices ◽  
Minimal Prime ◽  
Lattice Homomorphisms ◽  
The Ideal

For two given finite lattices $L$ and $M$, we introduce the ideal of lattice homomorphism $J(L,M)$, whose minimal monomial generators correspond to lattice homomorphisms $\phi : L\to M$. We show that $L$ is a distributive lattice if and only if the equidimensinal part of $J(L,M)$ is the same as the equidimensional part of the ideal of poset homomorphisms $I(L,M)$. Next, we study the minimal primary decomposition of $J(L,M)$ when $L$ is a distributive lattice and $M=[2]$. We present some methods to check if a monomial prime ideal belongs to $\mathrm{ass}(J(L,[2]))$, and we give an upper bound in terms of combinatorial properties of $L$ for the height of the minimal primes. We also show that if each minimal prime ideal of $J(L,[2])$ has height at most three, then $L$ is a planar lattice and $\mathrm{width}(L)\leq 2$. Finally, we compute the minimal primary decomposition when $L=[m]\times [n]$ and $M=[2]$.

The Quotient Problem for Noetherian Rings

1981 ◽  
Vol 33 (3) ◽  
pp. 734-748 ◽  
Author(s):  
Bruno J. Müller
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Noetherian Rings ◽  
Associated Primes ◽  
Singular Proposition ◽  
Role Taking ◽  
Composition Factors

Our work was motivated by attempts to find a criterion for the existence of a classical quotient ring, for a noetherian ring, in analogy with the various known criteria for the existence of an artinian classical quotient ring ([9], [10], [13], [2]).We have restricted our attention to Krull symmetric noetherian rings R, and we make heavy use of the fact that all their Krull composition factors are non-singular (Proposition 7). The collection Kprime R of the associated primes of the Krull composition factors of R plays a central role, taking the place of the collection of the associated primes of R.

scholarly journals Completely pseudo-valuation rings and their extensions

2014 ◽  
Vol 95 (109) ◽  
pp. 249-254
Author(s):  
Vijay Bhat
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Valuation Ring ◽  
Rational Numbers ◽  
Commutative Noetherian Ring ◽  
Valuation Rings ◽  
Minimal Prime

Recall that a commutative ring R is said to be a pseudo-valuation ring if every prime ideal of R is strongly prime. We define a completely pseudovaluation ring. Let R be a ring (not necessarily commutative). We say that R is a completely pseudo-valuation ring if every prime ideal of R is completely prime. With this we prove that if R is a commutative Noetherian ring, which is also an algebra over Q (the field of rational numbers) and ? a derivation of R, then R is a completely pseudo-valuation ring implies that R[x, ?] is a completely pseudo-valuation ring. We prove a similar result when prime is replaced by minimal prime.

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