• East Asian mathematical journal
• /
• 제17권2호
• /
• pp.367-374
• /
• 2001

Northcott and McKerrow proved that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-1}]$ is an injective left R[x]-module. Park generalized Northcott and McKerrow's result so that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-S}]$ is an injective left $R[x^S]$-module, where S is a submonoid of $\mathbb{N}$($\mathbb{N}$ is the set of all natural numbers). In this paper we extend the injective property to the Laurent power series module so that if R is a ring and E is an injective left R-module, then $E[[x^{-1},x]]$ is an injective left $R[x^S]$-module.

• 대한수학회보
• /
• 제58권1호
• /
• pp.217-224
• /
• 2021

In this paper, we study Ding injective modules over Frobenius extensions. Let R ⊂ A be a separable Frobenius extension of rings and M any left A-module, it is proved that M is a Ding injective left A-module if and only if M is a Ding injective left R-module if and only if A ⊗R M (HomR(A, M)) is a Ding injective left A-module.

• 대한수학회논문집
• /
• 제10권4호
• /
• pp.811-813
• /
• 1995

The purpose of this paper is to prove the divisibility of a direct injective module and every closed submodule of a $\pi$-injective module M is a direct summand of M.

• 대한수학회보
• /
• 제57권4호
• /
• pp.973-990
• /
• 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.

• 대한수학회지
• /
• 제50권5호
• /
• pp.1051-1066
• /
• 2013

By utilizing known characterizations of ${\omega}$-Noetherian rings in terms of injective modules, we give more characterizations of ${\omega}$-Noetherian rings. More precisely, we show that a commutative ring R is ${\omega}$-Noetherian if and only if the direct limit of GV -torsion-free injective R-modules is injective; if and only if every R-module has a GV -torsion-free injective (pre)cover; if and only if the direct sum of injective envelopes of ${\omega}$-simple R-modules is injective; if and only if the essential extension of the direct sum of GV -torsion-free injective R-modules is the direct sum of GV -torsion-free injective R-modules; if and only if every $\mathfrak{F}_{w,f}(R)$-injective ${\omega}$-module is injective; if and only if every GV-torsion-free R-module admits an $i$-decomposition.

• 대한수학회보
• /
• 제50권2호
• /
• pp.459-467
• /
• 2013

In this paper, Matlis injective modules are introduced and studied. It is shown that every R-module has a (special) Matlis injective preenvelope over any ring R and every right R-module has a Matlis injective envelope when R is a right Noetherian ring. Moreover, it is shown that every right R-module has an ${\mathcal{F}}^{{\perp}1}$-envelope when R is a right Noetherian ring and $\mathcal{F}$ is a class of injective right R-modules.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제8권2호
• /
• pp.175-183
• /
• 2001

The purpose of this paper is to find the necessary find sufficient conditions for a module to be a direct injective module. Moreover, we focus on the possibility that a direct injective module can be related with arbitrary module and Hom functor like an injective module.

• 대한수학회논문집
• /
• 제15권2호
• /
• pp.257-261
• /
• 2000

Northcott and Mckerrow proved that if R is a left noe-therian ring and E is an injective left R-module, then E[x-1] is an injective left R[x]-module. In this paper we generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an in-jective left R-module, then E[x-S] is an injective left R[xS]-module, where S is a submonoid of N (N is the set of all natural numbers).

• Korean Journal of Mathematics
• /
• 제25권4호
• /
• pp.587-606
• /
• 2017

This paper is a continuation of study semi-potentness endomorphism rings of module. We give some other characterizations of endomorphism ring to be semi-potent. New results are obtained including necessary and sufficient conditions for the endomorphism ring of semi(injective) projective module to be semi-potent. Finally, we characterize a module M whose endomorphism ring it is semi-potent via direct(injective) projective modules. Several properties of the endomorphism ring of a semi(injective) projective module are obtained. Besides to that, many necessary and sufficient conditions are obtained for semi-projective, semi-injective modules to be semi-potent and co-semi-potent modules.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제6권1호
• /
• pp.9-12
• /
• 1999

The purpose of this paper is to show that by the divisibility of a direct injective module, we obtain some results with respect to a direct injective module.