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Papers by Ajim Uddin Ansari

Research paper thumbnail of S-Primary Decomposition in S-Noetherian Rings

arXiv (Cornell University), Dec 31, 2023

Research paper thumbnail of Existence and uniqueness of <i>S</i> -primary decomposition in <i>S</i> -Noetherian modules

Existence and uniqueness of S -primary decomposition in S -Noetherian modules

Communications in algebra, May 15, 2024

Research paper thumbnail of Graded Prime Ideals Attached to a Group Graded Module

Iranian Journal of Mathematical Sciences and Informatics, Sep 1, 2022

Let G be a finitely generated abelian group and M be a G-graded A-module. In general, G-associate... more Let G be a finitely generated abelian group and M be a G-graded A-module. In general, G-associated prime ideals to M may not exist. In this paper, we introduce the concept of G-attached prime ideals to M as a generalization of G-associated prime ideals which gives a connection between certain G-prime ideals and G-graded modules over a (not necessarily G-graded Noetherian) ring. We prove that the Gattached prime ideals exist for every nonzero G-graded module and this generalization is proper. We transfer many results of G-associated prime ideals to G-attached prime ideals and give some applications of it.

Research paper thumbnail of Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

International Electronic Journal of Algebra, 2020

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer doma... more In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.

Research paper thumbnail of Graded S-Noetherian Modules

International Electronic Journal of Algebra

Let GGG be an abelian group and SSS a given multiplicatively closed subset of a commutative GGG-g... more Let GGG be an abelian group and SSS a given multiplicatively closed subset of a commutative GGG-graded ring AAA consisting of homogeneous elements. In this paper, we introduce and study GGG-graded SSS-Noetherian modules which are a generalization of SSS-Noetherian modules. We characterize GGG-graded SSS-Noetherian modules in terms of SSS-Noetherian modules. For instance, a GGG-graded AAA-module MMM is GGG-graded SSS-Noetherian if and only if MMM is SSS-Noetherian, provided GGG is finitely generated and SSS is countable. Also, we generalize some results on GGG-graded Noetherian rings and modules to GGG-graded SSS-Noetherian rings and modules.

Research paper thumbnail of Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

International Electronic Journal of Algebra, 2020

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer doma... more In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.

Research paper thumbnail of DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN

International Electronic Journal of Algebra, 2020

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer doma... more In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.

Research paper thumbnail of S-Primary Decomposition in S-Noetherian Rings

arXiv (Cornell University), Dec 31, 2023

Research paper thumbnail of Existence and uniqueness of <i>S</i> -primary decomposition in <i>S</i> -Noetherian modules

Existence and uniqueness of S -primary decomposition in S -Noetherian modules

Communications in algebra, May 15, 2024

Research paper thumbnail of Graded Prime Ideals Attached to a Group Graded Module

Iranian Journal of Mathematical Sciences and Informatics, Sep 1, 2022

Let G be a finitely generated abelian group and M be a G-graded A-module. In general, G-associate... more Let G be a finitely generated abelian group and M be a G-graded A-module. In general, G-associated prime ideals to M may not exist. In this paper, we introduce the concept of G-attached prime ideals to M as a generalization of G-associated prime ideals which gives a connection between certain G-prime ideals and G-graded modules over a (not necessarily G-graded Noetherian) ring. We prove that the Gattached prime ideals exist for every nonzero G-graded module and this generalization is proper. We transfer many results of G-associated prime ideals to G-attached prime ideals and give some applications of it.

Research paper thumbnail of Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

International Electronic Journal of Algebra, 2020

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer doma... more In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.

Research paper thumbnail of Graded S-Noetherian Modules

International Electronic Journal of Algebra

Let GGG be an abelian group and SSS a given multiplicatively closed subset of a commutative GGG-g... more Let GGG be an abelian group and SSS a given multiplicatively closed subset of a commutative GGG-graded ring AAA consisting of homogeneous elements. In this paper, we introduce and study GGG-graded SSS-Noetherian modules which are a generalization of SSS-Noetherian modules. We characterize GGG-graded SSS-Noetherian modules in terms of SSS-Noetherian modules. For instance, a GGG-graded AAA-module MMM is GGG-graded SSS-Noetherian if and only if MMM is SSS-Noetherian, provided GGG is finitely generated and SSS is countable. Also, we generalize some results on GGG-graded Noetherian rings and modules to GGG-graded SSS-Noetherian rings and modules.

Research paper thumbnail of Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

International Electronic Journal of Algebra, 2020

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer doma... more In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.

Research paper thumbnail of DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN

International Electronic Journal of Algebra, 2020

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer doma... more In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.