Prime Submodules and Local Gabriel Correspondence in σ[<i>M</i>]

Abstract. Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M -prime (resp. M -semiprime) submodule of X such that in the case M = R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M -m-system sets, M -n-system sets, M -prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M , called "prime M -ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and Hom R (M, X) = 0 for all modules X in the category σ[M ], then there is a one-to-one correspondence between isomorphism classes of indecomposable M -injective modules in σ[M ] and prime M -ideals of M . Also, we investigate the prime M -ideals, M -prime submodules and M -prime radical of Artinian modules.

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“…We note that, since in Lemma 2.29 we assume that M is projective in σ[M ], so our product coincides with the product defined in [6 …”

Section: Proof (⇒) Letmentioning

confidence: 99%

Prime M-Ideals, M-Prime Submodules, M-Prime Radical and M-Baer's Lower Nilradical of Modules

Beachy¹,

Behboodi²,

Yazdi³

2013

Journal of the Korean Mathematical Society

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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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