• Kyungpook Mathematical Journal
• /
• v.49 no.2
• /
• pp.255-263
• /
• 2009

For an endomorphism ${\alpha}$ of R, in [1], a module $M_R$ is called ${\alpha}$-compatible if, for any $m{\in}M$ and $a{\in}R$, ma = 0 iff $m{\alpha}(a)$ = 0, which are a generalization of ${\alpha}$-reduced modules. We study on the relationship between the quasi-Baerness and p.q.-Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [2] and some results in [9]. In particular, we show: for an ${\alpha}$-compatible module $M_R$ (1) $M_R$ is p.q.-Baer module iff $M[x;{\alpha}]_{R[x;{\alpha}]}$ is p.q.-Baer module. (2) for an automorphism ${\alpha}$ of R, $M_R$ is p.q.-Baer module iff $M[x,x^{-1};{\alpha}]_{R[x,x^{-1};{\alpha}]}$ is p.q.-Baer module.

• Kyungpook Mathematical Journal
• /
• v.58 no.3
• /
• pp.463-472
• /
• 2018

A Q-module is a module in which every nonzero submodule of M is a finite product of primary submodules of M. This paper is devoted to study some properties of Dedekind modules and Q-modules.

• Korean Journal of Mathematics
• /
• v.18 no.3
• /
• pp.243-252
• /
• 2010

In this paper we extend the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left R-modules to the projective properties of representations of quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\rightarrow}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. And we show $0{\rightarrow}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module if and only if $0{\rightarrow}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. Then we show if P is a projective left R-module then $R[x]\longrightarrow^{id}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. We also show that if $L\longrightarrow^{id}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module, then $L[x]\longrightarrow^{id}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.3
• /
• pp.445-456
• /
• 2008

Let M be a right R-module, G an ordered group and ${\sigma}$ a map from G into the group of automorphisms of R. The conditions under which the Malcev-Neumann module M* ((G)) is a PS module and a p.q.Baer module are investigated in this paper. It is shown that: (1) If $M_R$ is a reduced ${\sigma}$-compatible module, then the Malcev-Neumann module M* ((G)) over a PS-module is also a PS-module; (2) If $M_R$ is a faithful ${\sigma}$-compatible module, then the Malcev-Neumann module M* ((G)) is a p.q.Baer module if and only if the right annihilator of any G-indexed family of cyclic submodules of M in R is generated by an idempotent of R.

• Kyungpook Mathematical Journal
• /
• v.47 no.3
• /
• pp.357-361
• /
• 2007

A right R-module P is $c$-separative provided that $$P{\oplus}P{{c}\atop{\simeq_-}}P{\oplus}Q{\Longrightarrow}P{\simeq_-}Q$$ for any right R-module Q. We get, in this paper, two sufficient conditions under which a right module is $c$-separative. A ring R is a hereditary ring provided that every ideal of R is projective. As an application, we prove that every projective right R-module over a hereditary ring is $c$-separative.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.6
• /
• pp.1905-1914
• /
• 2013

Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and $N_v=\{f{\in}D[X]|c(f)_v=D\}$. Let $q(M)=M{\otimes}_D\;K$ and $M_{w_D}$={$x{\in}q(M)|xJ{\subseteq}M$ for a nonzero finitely generated ideal J of D with $J_v$ = D}. In this paper, we show that $M_{w_D}=M[X]_{N_v}{\cap}q(M)$ and $(M[X])_{w_{D[X]}}{\cap}q(M)[X]=M_{w_D}[X]=M[X]_{N_v}{\cap}q(M)[X]$. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]-module if and only if $M[X]_{N_v}$ is a Noetherian $D[X]_{N_v}$-module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if $D[X]_{N_v}$ is a Noetherian domain.

• Korean Journal of Mathematics
• /
• v.20 no.3
• /
• pp.343-352
• /
• 2012

In this paper we show that the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\longrightarrow}0{\longrightarrow}P[x]$ is a projective representation of a quiver Q as $R[x]$-modules, but $P[x]{\longrightarrow}0{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. And we show a representation $0{\longrightarrow}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module, but $P[x]\longrightarrow^{id}P[x]{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. Then we show a representation $P[x]\longrightarrow^{id}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1563-1567
• /
• 2021

We show that if M is a Utumi module, in particular if M is quasi-continuous, then M = Q ⊕ K, where Q is quasi-injective that is both a square-full as well as a dual-square-full module, K is a square-free module, and Q & K are orthogonal. Dually, we also show that if M is a dual-Utumi module whose local summands are summands, in particular if M is quasi-discrete, then M = P ⊕ K where P is quasi-projective that is both a square-full as well as a dual-square-full module, K is a dual-square-free module, and P & K are factor-orthogonal.

• Korean Journal of Mathematics
• /
• v.17 no.3
• /
• pp.271-281
• /
• 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 $M_2$.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
• /
• v.13 no.5
• /
• pp.348-355
• /
• 2001

The coupled conduction and convection heat transfer from a protruding heated module in a horizontal channel with a variation of channel height is experimentally investigated. The input power to the module is 3, 7W and thermal resistance of module support is 0.06 , 1.03 and 158K/W. the Reynolds number ranged from 350 to 4500 corresponding to the inlet velocity(0.4~1.3 m/s) and channel height(11~35 mm). The results were obtained that the decrease of thermal resistance of module support reduces the module temperature by redistributing the heat flux and the overall thermal resistance of the module. In the study the effect of channel height is very significant in the adiabatic condition, but negligible in the conjugate condition. Finally, correlations for Nusselt number and $Q_B$/Q with a variation of Reynolds number were developed respectively.