• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.771-778
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• 2005

In this paper we shall give some group properties derived from the characteristic ring-module $_X(M)$, using the fact that $_X(M)_H$ is a conjugate to $_X(M)_{Ha}$ when M is an invertible right R-module. Also we shall prove that_X(M)$ is group-isomorphic to TR and some normal subgroup properties if M is invertible and R is commutative.

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

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.559-571
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• 2017

We investigate modules M having the injective property relative to nonsingular modules. Such modules are called "$\mathcal{N}$-injective modules". It is shown that M is an $\mathcal{N}$-injective R-module if and only if the annihilator of $Z_2(R_R)$ in M is equal to the annihilator of $Z_2(R_R)$ in E(M). Every $\mathcal{N}$-injective R-module is injective precisely when R is a right nonsingular ring. We prove that the endomorphism ring of an $\mathcal{N}$-injective module has a von Neumann regular factor ring. Every (finitely generated, cyclic, free) R-module is $\mathcal{N}$-injective, if and only if $R^{(\mathbb{N})}$ is $\mathcal{N}$-injective, if and only if R is right t-semisimple. The $\mathcal{N}$-injective property is characterized for right extending rings, semilocal rings and rings of finite reduced rank. Using the $\mathcal{N}$-injective property, we determine the rings whose all nonsingular cyclic modules are injective.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.215-218
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• 1999

We investigate characterizations of rings whose simple singular right R-modules are GP-injective. It is proved that if R is a semiprime ring whose simple singular right R-modules are GP-injective, then the center $Z(R)$ of R is a von Neumann regular ring. We consider the condition ($^*$): R satisfies $l(a){\subseteq}r(a)$ for any $a{\in}R$. Also it is shown that if R satisfies ($^*$) and every simple singular right R-module is GP-injective, then R is a reduced weakly regular ring.