• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.1083-1096
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• 2012

For a given ideal I of a Noetherian ring R and an arbitrary integer ${\kappa}{\geq}-1$, we apply the concept of ${\kappa}$-regular sequences and the notion of ${\kappa}$-depth to give some results on modules called ${\kappa}$-Cohen Macaulay modules, which in local case, is exactly the ${\kappa}$-modules (as a generalization of f-modules). Meanwhile, we give an expression of local cohomology with respect to any ${\kappa}$-regular sequence in I, in a particular case. We prove that the dimension of homology modules of the Koszul complex with respect to any ${\kappa}$-regular sequence is at most ${\kappa}$. Therefore homology modules of the Koszul complex with respect to any filter regular sequence has finite length.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.253-266
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• 2014

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.225-231
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• 2007

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal$({\phi})$. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope $E[x^{-1}]$ of an inverse polynomial module $M[x^{-1}]$ as a left R[x]-module and we can define an associative Galois group Gal$({\phi}[x^{-1}])$. In this paper we describe the relations between Gal$({\phi})$ and Gal$({\phi}[x^{-1}])$. Then we extend the Galois group of inverse polynomial module and can get Gal$({\phi}[x^{-s}])$, where S is a submonoid of $\mathbb{N}$ (the set of all natural numbers).

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.711-719
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• 2018

In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring R, if its maximal right quotient ring has semisimple dimension as a right R-module, then R is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right R-modules have semisimple dimension if and only if the free right R-module ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.989-996
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• 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper we study the amalgamated duplication ring $R{\bowtie}I$ which is introduced by D'Anna and Fontana. It is shown that if R is generically Cohen-Macaulay (resp. generically Gorenstein) and I is generically maximal Cohen-Macaulay (resp. generically canonical module), then $R{\bowtie}I$ is generically Cohen-Macaulay (resp. generically Gorenstein). We also de ned generically quasi-Gorenstein ring and we investigate when $R{\bowtie}I$ is generically quasi-Gorenstein. In addition, it is shown that $R{\bowtie}I$ is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if R is approximately Gorenstein, then $R{\bowtie}I$ is approximately Gorenstein.

• Journal of the Korean Mathematical Society
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• v.51 no.5
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• pp.971-985
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• 2014

All modules considered in this note are over associative commutative rings with an identity element. We show that a ${\omega}$-local module M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that ${\omega}$-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated amply Rad-supplemented left modules over any ring (not necessarily commutative).

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.289-298
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• 2017

Let R denote a commutative Noetherian (not necessarily local) ring and let I be an ideal of R of dimension one. The main purpose of this note is to show that the category ${\mathfrak{M}}(R,\;I)_{com}$ of I-cominimax R-modules forms an Abelian subcategory of the category of all R-modules. This assertion is a generalization of the main result of Melkersson in [15]. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category ${\mathcal{C}}^1_B(R)$ of all R-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all R-modules.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.319-329
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• 2017

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1331-1336
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• 2017

Assume that R is a commutative Noetherian ring with non-zero identity, a is an ideal of R, X is an R-module, and t is a non-negative integer. In this paper, we present upper bounds for the injective dimension of X in terms of the injective dimensions of its local cohomology modules and an upper bound for the injective dimension of $H^t_{\alpha}(X)$ in terms of the injective dimensions of the modules $H^i_{\alpha}(X)$, $i{\neq}t$, and that of X. As a consequence, we observe that R is Gorenstein whenever $H^t_{\alpha}(R)$ is of finite injective dimension for all i.

• East Asian mathematical journal
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• v.24 no.2
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• pp.139-144
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• 2008

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal($\phi$). Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope E[$x^{-1}$] of an inverse polynomial module M[$x^{-1}$] as a left R[x]-module and we can define an associative Galois group Gal(${\phi}[x^{-1}]$). In this paper we extend the Galois group of inverse polynomial module and can get Gal(${\phi}[x^{-s}]$), where S is a submonoid of $\mathds{N}$ (the set of all natural numbers).