Krull Dimension and Classical Krull Dimension of Modules

Pérez, Jaime Castro; Montes, José Rı́os

doi:10.1080/00927872.2013.781611

Cited by 11 publications

(3 citation statements)

References 13 publications

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“…semiprime) submodule. Many properties known for prime or semiprime ideals can be extended to prime or semiprime sumodules as it has been done in [1,[5][6][7]13,14]. Recall that a ring R is left Goldie, if R satisfies the ascending chain condition (acc) on left annihilators and has finite uniform dimension.…”

Section: Preliminariesmentioning

confidence: 99%

The nilpotency of the prime radical of a Goldie module

Beachy¹,

Bárcenas²

2021

Preprint

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Section: Preliminariesmentioning

confidence: 99%

The nilpotency of the prime radical of a Goldie module

Beachy¹,

Bárcenas²

2021

Preprint

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“…It follows from[7, Proposition 3.4] that M is non M -singular.Remark 3.6. The proof (a)⇒(b) is almost the same as that in[7, Proposition 3.13] but the sentence of that Proposition is written with different hypotheses. Also, note that the proof (a)⇒(b) shows that M is essentially compressible in the sense of[13].…”

mentioning

confidence: 99%

“…(b)⇒(a) Since M is projective in σ[M ] and semiprime, then M is retractable. It follows from[7, Proposition 3.4] that M is non M -singular.Remark 3.6. The proof (a)⇒(b) is almost the same as that in[7, Proposition 3.13] but the sentence of that Proposition is written with different hypotheses.…”

mentioning

confidence: 99%

Reduced rank in $σ[M]$

Beachy¹,

Bárcenas²

2022

Preprint

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Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context. We study the quotient category of σ[M ] modulo the hereditary torsion theory cogenerated by the M -injective hull of M when M is a semiprime Goldie module. We prove that this quotient category is spectral. Later we consider the hereditary torsion theory in σ[M ] cogenerated by the M -injective hull of M/L(M ) where L(M ) is the prime radical of M , and we characterize when the module of quotients of M , respect to this torsion theory, has finite length in the quotient category. At the end we give conditions on a module M with endomorphism ring S in order to get that S is an order in an Artininan ring, extending Small's Theorem.

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On the De Morgan’s Laws for Modules

Bárcenas

Sandoval-Miranda

Zaldívar-Corichi

2021

Appl Categor Struct

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In this investigation we give a module-theoretic counterpart of the well known De Morgan's laws for rings and topological spaces. We observe that the module-theoretic De Morgan's laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan's laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law.

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Krull Dimension and Classical Krull Dimension of Modules

Cited by 11 publications

References 13 publications

The nilpotency of the prime radical of a Goldie module

The nilpotency of the prime radical of a Goldie module

Reduced rank in $σ[M]$

On the De Morgan’s Laws for Modules

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