A note on the countable union of prime submodules

Pournaki, M. R.; Tousi, Massoud

doi:10.1155/s0161171201011085

Cited by 2 publications

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“…There are also the Sharp-Vamos prime avoidance theorems for noetherian complete local rings for countable prime ideals, stated as 16.7 A and 16.7 B in [4], see also [13], [10]. We have extended the latter result to complete noetherian semi-local rings in [8], for some purpose.…”

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confidence: 87%

The Prime Avoidance Lemma Revisited

Karamzadeh¹

2012

Kyungpook mathematical journal

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Abstract. We show that the above lemma and its well-known refinement are valid, in a general setting, in non-commutative rings. Some interesting consequences are also observed.The above lemma is in every single text book on commutative algebra, but not even a single book on non-commutative algebra, as yet (except [12, Proposition 2.12.7], where the validity of a special variant of this lemma is shown), proves it for non-commutative rings. We recall that, in the proof of [7, Proposition 2], which is in fact the proof of the converse of the Generalized Principal Ideal Theorem of Krull for non-commutative rings, see also [5, Theorem 153], we invoked the lemma for the non-commutative case, even without mentioning it, and gave no proof for it ( note, it seems (at least to us) that the lemma for non-commutative rings was overlooked in the literature, at that time, and although I knew of a proof for a generalization of this lemma then, but did not present it in [7], on purpose, for the reason that we will see, shortly). Let R be a ring with identity, if S is a subring of R without containing the identity of R, we say that S is a subring−1 (see [4, Chapter 16]). The above lemma states that, if P 1 , P 2 , . . . , P n are ideals of a commutative ring R with identity such that at most two of the P i 's are not prime and S is a subring−1 of R contained in P 1 ∪ P 2 ∪ . . . ∪ P n , then S ⊆ P k for some P k (note, in most cases S is an ideal ). Without any doubt, as it is rightly mentioned in [13], the Prime Avoidance Lemma is one of the fundamental cornerstones of commutative algebra. Its numerous applications in the field, makes one surely claim that no one working in the commutative algebra can do without it. The lemma for commutative rings goes back to a 1957 paper by McCoy, and Kaplansky generalized it in his 1974 book on commutative rings, see [4], [5], respectively. The reader might consult ([4, Chapter 16]), [9] and [11]) for a short history and some variations of this lemma. There is also a very useful refinement of the lemma, which says that whenever P 1 , P 2 , . . . , P n are prime ideals of the commutative ring R and I is an ideal of R, a ∈ R with

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confidence: 87%

The Prime Avoidance Lemma Revisited

Karamzadeh¹

2012

Kyungpook mathematical journal

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“…Details about -dim and width can be found in Roberts [2], Kirby [3], and Ooishi [4]; there is a general fact: for any Artinian -module width ≤ -dim < ∞ holds and is co-Cohen-Macaulay if and only if width = -dim holds (cf. [5][6][7]). Tang [8] has shown that if either ≤ 2 or is Cohen-Macaulay, then m ( ) is co-Cohen-Macaulay (see also [9]).…”

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confidence: 99%

Co-Cohen-Macaulay Modules and Local Cohomology

Saremi

Mafi

2013

Journal of Mathematics

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Let (, m) be a commutative Noetherian local ring and let be a finitely generated-module of dimension. Then the following statements hold: (a) if width (m ()) ≥ −1 for all with 2 ≤ < , then m () is co-Cohen-Macaulay of Noetherian dimension ; (b) if is an unmixed-module and depth ≥ − 1, then m () is co-Cohen-Macaulay of Noetherian dimension if and only if −1 m () is either zero or co-Cohen-Macaulay of Noetherian dimension − 2. As consequence, if m () is co-Cohen-Macaulay of Noetherian dimension for all with 0 ≤ < , then m () is co-Cohen-Macaulay of Noetherian dimension .

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A note on the countable union of prime submodules

Cited by 2 publications

References 3 publications

The Prime Avoidance Lemma Revisited

The Prime Avoidance Lemma Revisited

Co-Cohen-Macaulay Modules and Local Cohomology

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