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RISS 인기검색어
- 검색키워드
- 저자 : Salahattin Ozdemir (검색결과 4 건)
- 무료
- 기관 내 무료
- 유료
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Ozdemir, Salahattin Korean Mathematical Society 2016 대한수학회지 Vol.53 No.2
Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is Rad-supplementing if and only if R is reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.
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Salahattin Ozdemir 대한수학회 2016 대한수학회지 Vol.53 No.2
Let $R$ be a ring, and let $M$ be a left $R$-module. If $M$ is Rad-supplementing, then every direct summand of $M$ is Rad-supplementing, but not each factor module of $M$. Any finite direct sum of Rad-supple\-menting modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. $M$ has a Rad-supplement in its injective envelope if and only if $M$ has a Rad-supplement in every essential extension. $R$ is left perfect if and only if $R$ is semilocal, reduced and the free left $R$-module $(_R R)^{(\mathbb{N})}$ is Rad-supplementing if and only if $R$ is reduced and the free left $R$-module $(_R R)^{(\mathbb{N})}$ is ample Rad-supplementing. $M$ is ample Rad-supplementing if and only if every submodule of $M$ is Rad-supplementing. Every left $R$-module is (ample) Rad-supplementing if and only if $R/P(R)$ is left perfect, where $P(R)$ is the sum of all left ideals $I$ of $R$ such that $\Rad I = I$.
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Yilmaz Durugun,Salahattin Ozdemir 대한수학회 2017 대한수학회지 Vol.54 No.4
A submodule $N$ of a module $M$ is called $\mathcal{S}$-closed (in $M$) if $M/N$ is nonsingular. It is well-known that the class $\closed$ of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-\closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class $\langle\mathcal{S}-\closed\rangle$ containing $\mathcal{S}-\closed$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring $R$, it coincides with the proper class generated by neat submodules if and only if $R$ is a right SI-ring. In abelian groups, the elements of this class are exactly torsion-splitting. In the second part, coprojective modules of this class which we call \emph{ec-flat} modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring $R$ of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class $\langle\mathcal{S}-\closed\rangle $ coincides with the class of pure-exact sequences of modules if and only if $R$ is a two-sided hereditary, two-sided $CS$-ring and every singular right module is a direct sum of finitely presented modules.
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Durgun, Yilmaz,Ozdemir, Salahattin Korean Mathematical Society 2017 대한수학회지 Vol.54 No.4
A submodule N of a module M is called ${\mathcal{S}}$-closed (in M) if M/N is nonsingular. It is well-known that the class Closed of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-Closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class ${\langle}\mathcal{S-Closed}{\rangle}$ containing $\mathcal{S-Closed}$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring R, it coincides with the proper class generated by neat submodules if and only if R is a right SI-ring. In abelian groups, the elements of this class are exactly torsionsplitting. In the second part, coprojective modules of this class which we call ec-flat modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring R of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class ${\langle}\mathcal{S-Closed}{\rangle}$ coincides with the class of pure-exact sequences of modules if and only if R is a two-sided hereditary, two-sided $\mathcal{CS}$-ring and every singular right module is a direct sum of finitely presented modules.
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