• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.693-699
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• 2010

In this paper we show that the flat property of a left R-module does not imply (carry over) to the corresponding inverse polynomial module. Then we define an induced inverse polynomial module as an R[x]-module, i.e., given an R-linear map f : M $\rightarrow$ N of left R-modules, we define $N+x^{-1}M[x^{-1}]$ as a left R[x]-module. Given an exact sequence of left R-modules $$0\;{\rightarrow}\;N\;{\rightarrow}\;E^0\;{\rightarrow}\;E^1\;{\rightarrow}\;0$$, where $E^0$, $E^1$ injective, we show $E^1\;+\;x^{-1}E^0[[x^{-1}]]$ is not an injective left R[x]-module, while $E^0[[x^{-1}]]$ is an injective left R[x]-module. Make a left R-module N as a left R[x]-module by xN = 0. We show inj $dim_R$ N = n implies inj $dim_{R[x]}$ N = n + 1 by using the induced inverse polynomial modules and their properties.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.609-616
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• 2005

In this paper we show that we can extend the purity of left R-modules to the case of polynomial modules, bipolynomial modules, and also inverse polynomial modules.

• 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).

• 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).

• Communications of the Korean Mathematical Society
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• v.15 no.2
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• pp.257-261
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• 2000

Northcott and Mckerrow proved that if R is a left noe-therian ring and E is an injective left R-module, then E[x-1] is an injective left R[x]-module. In this paper we generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an in-jective left R-module, then E[x-S] is an injective left R[xS]-module, where S is a submonoid of N (N is the set of all natural numbers).

• East Asian mathematical journal
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• v.21 no.1
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• pp.105-112
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• 2005

In this paper we show that we can extend the purity extension properties of left R-modules to the various generalized inverse polynomial modules.

• Honam Mathematical Journal
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• v.28 no.3
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• pp.377-386
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• 2006

For a commutative ring with unity R, it is proved that R is a PF-ring if and only if the annihilator, $ann_R(a)$, for each $a{\in}R$ is a pure ideal in R. Also it is proved that the polynomial ring, R[x], is a PF-ring if and only if R is a PF-ring. Finally, we prove that M as an R-module is PF-module if and only if M[x] is a PF R[x]-module. Also M is a PP R-module if and only if M[x] is a PP R[x]-module.

• Kyungpook Mathematical Journal
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• v.45 no.1
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• pp.131-135
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• 2005

Let an integral domain R be locally polynomial over an integral domain D and let R be a content module over D. We show that if D is a PVMD, then $$Cl_t(R){\sim_=}Cl_t(D)$$. This generalizes the polynomial case. We also show that R is a PVMD if and only if D is a PVMD if and only if $R_{N_v}$ is a PVMD.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.23-33
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• 2011

Extending the notion of McCoy rings, we introduce the class of McCoy modules. Over a given ring R, it contains the class of Armendariz modules (over R). Some properties of this class of modules are established, and equivalent conditions for McCoy modules are given. Moreover, we study the relationship between a module and its polynomial module. Several known results relating to McCoy rings can be obtained as corollaries of our results.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1067-1079
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• 2012

Let M be a right R-module, (S, ${\leq}$) a strictly totally ordered monoid which is also artinian and ${\omega}:S{\rightarrow}Aut(R)$ a monoid homomorphism, and let $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ denote the generalized inverse polynomial module over the skew generalized power series ring [[$R^{S,{\leq}},{\omega}$]]. In this paper, we prove that $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same uniform dimension as its coefficient module $M_R$, and that if, in addition, R is a right perfect ring and S is a chain monoid, then $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same couniform dimension as its coefficient module $M_R$.