• 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.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.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).

• 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.

• 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).

• 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 $M_R$.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.45-55
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• 1998

• East Asian mathematical journal
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• v.16 no.1
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• pp.111-123
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• 2000

Northcott and McKerrow proved that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-1}]$ is an injective left R[xl-module. Park generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-S}]$ is an injective left $R[x^s]$-module, where S is a submonoid of N(N is the set of all natural numbers). In this paper we show $$Hom_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Hom_R(M,\;N)[[x^S]]$$ and using the above result and this isomorphism, finally we show that $$Ext^i_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Ext^i_R(M,\;N)[[x^S]]$$.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.307-314
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• 2012

Let M be a left R-module and R be a ring with unity, and $S=\{0,2,3,4,{\ldots}\}$ be a submonoid. Then $M[x^{-s}]=\{a_0+a_2x^{-2}+a_3x^{-3}+{\cdots}+a_nx^{-n}{\mid}a_i{\in}M\}$ is an $R[x^s]$-module. In this paper we show some properties of $M[x^{-s}]$ as an $R[x^s]$-module. Let $f:M{\rightarrow}N$ be an R-linear map and $\overline{M}[x^{-s}]=\{a_2x^{-2}+a_3x^{-3}+{\cdots}+a_nx^{-n}{\mid}a_i{\in}M\}$ and define $N+\overline{M}[x^{-s}]=\{b_0+a_2x^{-2}+a_3x^{-3}+{\cdots}+a_nx^{-n}{\mid}b_0{\in}N,\;a_i{\in}M}$. Then $N+\overline{M}[x^{-s}]$ is an $R[x^s]$-module. We show that given a short exact sequence $0{\rightarrow}L{\rightarrow}M{\rightarrow}N{\rightarrow}0$ of R-modules, $0{\rightarrow}L{\rightarrow}M[x^{-s}]{\rightarrow}N+\overline{M}[x^{-s}]{\rightarrow}0$ is a short exact sequence of $R[x^s]$-module. Then we show $E_1+\overline{E_0}[x^{-s}]$ is not an injective left $R[x^s]$-module, in general.