• The Mathematical Education
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• v.23 no.1
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• pp.25-26
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• 1984

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.177-185
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• 2011

A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I ${\neq}$ = R. A right R-module M is called PS-injective if every R-homomorphism f : aR ${\rightarrow}$ M for every principally small right ideal aR can be extended to R ${\rightarrow}$ M. A ring R is called right PS-injective if R is PS-injective as a right R-module. We develop, in this article, PS-injectivity as a generalization of P-injectivity and small injectivity. Many characterizations of right PS-injective rings are studied. In light of these facts, we get several new properties of a right GPF ring and a semiprimitive ring in terms of right PS-injectivity. Related examples are given as well.

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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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.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.225-237
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• 2019

We introduce a larger ideal ${\mathcal{N}}_{wp}$ of the ideal of p-nuclear operators. We obtain isometric representations of the injective and surjective hulls of ${\mathcal{N}}_{wp}$ and study them.

• The Mathematical Education
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• v.31 no.2
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• pp.141-144
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• 1992

In this paper, some properties of p-injective ring is studied: The Jacobson radical of a pinjective ring which satisfies the ascending chain condition on essential left ideals is nilpotent. Also, the left singular ideal of a ring which satisfies the ascending chain condition on essential left ideals is nilpotent. Finally, we give an example which shows that a semiprime left p-injective ring such that every essential left ideal is two-sided is not necessarily to be strongly regular.egular.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.179-186
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• 2019

This article mainly concentrates on the open question whether a right self-injective ring R is necessary QF if $R/S_l$ is left Goldie. It is answered affirmatively under the condition $S_l{\subseteq}S_r$, where $S_l$ and $S_r$ denote the left socle and right socle of R respectively. And the original condition "right self-injective" can be weakened to "right CS and right P-injective". It is also proved that a semiperfect, left and right mininjective ring R is QF if $S_r{\subseteq}^{ess}$ $R_R$ and $R/S_l$ is left Goldie.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.343-352
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• 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
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• v.49 no.2
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• pp.395-409
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• 2012

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let ${\pi}$ be a group and let M ${\rightarrow}$ N be a homomorphism between projective $\mathbb{Z}[{\pi}]$-modules such that $\mathbb{Z}_p\;{\otimes}_{\mathbb{Z}[{\pi}]}M{\rightarrow}\mathbb{Z}_p{\otimes}_{\mathbb{Z}[{\pi}]}\;N$ is injective; for which other right $\mathbb{Z}[{\pi}]$-modules V is the induced map $V{\otimes}_{\mathbb{Z}[{\pi}]}\;M{\rightarrow}\;V{\otimes}_{\mathbb{Z}[{\pi}]}\;N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.

• Journal of the Korean Mathematical Society
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• v.47 no.4
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• pp.691-704
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• 2010

Over a ring R, let $P_R$ be a finitely generated projective right R-module. Then we define the G-injector (G-projector) if $P_R$ preservers Gorenstein injective modules (Gorenstein projective modules), the Gflator if $P_R$ preservers Gorenstein flat modules. G-injector (G-flator) and G-injector are characterized focus primarily on the cases where R is a Gorenstein ring, and under this condition we also study the relations between the injector (projector, flator) and the G-injector (G-projector, G-flator). Over any ring we also give the characteristics of G-injector (G-flator) by the Gorenstein injective (Gorenstein flat) dimensions of modules.

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