• 대한수학회보
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• 제43권4호
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• pp.755-762
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• 2006

For a commutative ring R and an injective cogenerator E in the category of R-modules, we characterize von Neumann regular rings and semisimple artinian rings in terms of the properties of dual modules with respect to E.

• 대한수학회논문집
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• 제21권1호
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• pp.37-43
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• 2006

We prove that $M_1\longrightarrow^f\;M_2$ is an injective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ if and only if $M_1\;and\;M_2$ are injective left R-modules, $M_1\longrightarrow^f\;M_2$ is isomorphic to a direct sum of representation of the types $E_l{\rightarrow}0$ and $M_1\longrightarrow^{id}\;M_2$ where $E_l\;and\;E_2$ are injective left R-modules. Then, we generalize the result so that a representation$M_1\longrightarrow^{f_1}\;M_2\; \longrightarrow^{f_2}\;\cdots\;\longrightarrow^{f_{n-1}}\;M_n$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\cdots}{\rightarrow}{\bullet}$ is an injective representation if and only if each $M_i$ is an injective left R-module and the representation is a direct sum of injective representations.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.83-94
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• 2016

Let M and N be modules over a ring R. The purpose of this paper is to study modules M, N for which the bi-module [M, N] is regular or pi. It is proved that the bi-module [M, N] is regular if and only if a module N is semi M-projective and $Im({\alpha}){\subseteq}^{\oplus}N$ for all ${\alpha}{\in}[M,N]$, if and only if a module M is semi N-injective and $Ker({\alpha}){\subseteq}^{\oplus}N$ for all ${\alpha}{\in}[M,N]$. Also, it is proved that the bi-module [M, N] is pi if and only if a module N is direct M-projective and for any ${\alpha}{\in}[M,N]$ there exists ${\beta}{\in}[M,N]$ such that $Im({\alpha}{\beta}){\subseteq}^{\oplus}N$, if and only if a module M is direct N-injective and for any ${\alpha}{\in}[M,N]$ there exists ${\beta}{\in}[M,N]$ such that $Ker({\beta}{\alpha}){\subseteq}^{\oplus}M$. The relationship between the Jacobson radical and the (co)singular ideal of [M, N] is described.

• 호남수학학술지
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• 제31권4호
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• pp.529-540
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• 2009

The authors have investigated several properties of endomorphism rings of modules. In particular, a closedly simple module and a closedly semi-simple module were defined and then the regularities of its rings of endomorphisms were discussed.

• 대한수학회보
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• 제37권1호
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• pp.1-19
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• 2000

We investigate in this paper the maximality of prime ideals in rings whose simple singular left R-modules are p-injective.

• 대한수학회지
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• 제43권1호
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• pp.65-76
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• 2006

Let $\tau$ be a hereditary torsion theory and let p be a prime ideal of a commutative ring R. We study the existence of (minimal) $\tau-injective$ submodules of the injective hull of R/p.

• 대한수학회논문집
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• 제25권4호
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• pp.497-510
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• 2010

Let R be a ring and n a fixed non-negative integer. $\cal{TI}_n$ (resp. $\cal{TF}_n$) denotes the class of all right R-modules of FGT-injective dimensions at most n (resp. all left R-modules of FGT-flat dimensions at most n). We prove that, if R is a right $\prod$-coherent ring, then every right R-module has a $\cal{TI}_n$-cover and every left R-module has a $\cal{TF}_n$-preenvelope. A right R-module M is called n-TI-injective in case $Ext^1$(N,M) = 0 for any $N\;{\in}\;\cal{TI}_n$. A left R-module F is said to be n-TI-flat if $Tor_1$(N, F) = 0 for any $N\;{\in}\;\cal{TI}_n$. Some properties of n-TI-injective and n-TI-flat modules and their relations with $\cal{TI}_n$-(pre)covers and $\cal{TF}_n$-preenvelopes are also studied.

• 대한수학회논문집
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• 제20권3호
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• pp.437-442
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• 2005

In studying injective covers, the modules C such that Hom(E, C) = 0 and $Ext^1$(E, C) = 0 for all injective module E play an important role because of Wakamatsu's lemma. If C is a module over the ring k[[x, y]] with k a field, the class of these modules C contains the class $\={D}$ of all direct summands of products of modules of finite length ([3, Theorem 2.9]). In this paper we show that every module over any commutative ring has a $\={D}$-preenvelope.

• 대한수학회지
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• 제51권3호
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• pp.509-525
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• 2014

Let R be a commutative ring with identity. An R-module M is said to be w-projective if $Ext\frac{1}{R}$(M,N) is GV-torsion for any torsion-free w-module N. In this paper, we define a ring R to be w-semi-hereditary if every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules and study some basic properties of w-injective modules. Using these concepts, we show that R is w-semi-hereditary if and only if the total quotient ring T(R) of R is a von Neumann regular ring and $R_m$ is a valuation domain for any maximal w-ideal m of R. It is also shown that a connected ring R is w-semi-hereditary if and only if R is a Pr$\ddot{u}$fer v-multiplication domain.

• 대한수학회보
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• 제56권1호
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• pp.103-110
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• 2019

In this paper, we give a characterization of prime submodules of an injective module over a Noetherian ring.