• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.203-218
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• 2013

Let R be a ring. A right R-module M is called GI-flat if $Tor^R_1(M,G)=0$ for every Gorenstein injective left R-module G. It is shown that GI-flat modules lie strictly between flat modules and copure flat modules. Suppose R is an $n$-FC ring, we prove that a finitely presented right R-module M is GI-flat if and only if M is a cokernel of a Gorenstein flat preenvelope K ${\rightarrow}$ F of a right R-module K with F flat. Then we study GI-flat dimensions of modules and rings. Various results in [6] are developed, some new characterizations of von Neumann regular rings are given.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.973-990
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• 2020

For the question "when is E(_RR) a flat left R-module for any ring R?", in this paper, we deal with a class of modules partaking in the hierarchy of injective and cotorsion modules, so-called Harmanci injective modules, which turn out by the motivation of relations among the concepts of injectivity, flatness and cotorsionness. We give some characterizations and properties of this class of modules. It is shown that the class of all Harmanci injective modules is enveloping, and forms a perfect cotorsion theory with the class of modules whose character modules are Matlis injective. For the objective we pursue, we characterize when the injective envelope of a ring as a module over itself is a flat module.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.439-456
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• 1994

Let $\tau$ be a given hereditary torsion theory for left R-module category R-Mod. The class of all $\tau$-torsion left R-modules, denoted by T is closed under homomorphic images, submodules, direct sums and extensions. And the class of all $\tau$-torsionfree left R-modules, denoted by $F$, is closed under submodules, injective hulls, direct products, and isomorphic copies ([3], Proposition 1.7 and 1.10).

• Communications of the Korean Mathematical Society
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• v.21 no.1
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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.

• Korean Journal of Mathematics
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• v.17 no.3
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• pp.271-281
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• 2009

We define injective and projective representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$. Then we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is projective if and only if each $M_1,\;M_2,\;M_3$ is projective left R-module and $f_1(M_1)$ is a summand of $M_2$ and $f_2(M_2)$ is a summand of $M_3$. And we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is injective if and only if each $M_1,\;M_2,\;M_3$ is injective left R-module and $ker(f_1)$ is a summand of $M_1$ and $ker(f_2)$ is a summand of $M_2$.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.865-868
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• 1997

Let R be a ring with a unity and let M be a unitary left R-module. In this paper, we establish [5, Proposition 2.8] by showing the proof of it. Moreover, from the above result, we obtain some properties of direct projective modules which have the summand sum property.

• East Asian mathematical journal
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• v.18 no.2
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• pp.245-259
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• 2002

In this paper, all rings or (left)near-rings R are associative, and for near-ring R, all R-groups are right R action and all modules are right R-modules. First, we begin with the study of rings in which all the additive endomorphisms or only the left multiplication endomorphisms are generated by ring endomorphisms and their properties. This study was motivated by the work on the Sullivan's Problem [14]. Next, for any right R-module M, we will introduce AM modules and investigate their basic properties. Finally, for any nearring R, we will also introduce MR-groups and study some of their properties.

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.1-6
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• 1994

In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.93-97
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• 1994

Previously T.Porter [3] has given a relative Chinese Remainder Theorem under the hypothesis that given ring R has at least one .tau.-closed maximal ideal (by his notation Ma $x_{\tau}$(R).neq..phi.). In this short paper we drop his overall hypothesis that Ma $x_{\tau}$(R).neq..phi. and give the proof and some related results with this Theorem. In this paper R will always denote a commutative ring with identity element and all modules will be unitary left R-modules unless otherwise specified. Let .tau. be a given hereditarty torsion theory for left R-module category R-Mod. The class of all .tau.-torsion left R-modules, dented by J is closed under homomorphic images, submodules, direct sums and extensions. And the class of all .tau.-torsionfree left R-modules, denoted by F, is closed under taking submodules, injective hulls, direct products, and isomorphic copies ([2], Proposition 1.7 and 1.10).

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1271-1290
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• 2013

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.