Finitely Generated Modules over Semilocal Rings and Characterizations of (Semi-)Perfect Rings

Chang, Chae-Hoon Department of Mathematics 2008 Kyungpook mathematical journal Vol.48 No.1

Lomp [9] has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sufficient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules.

On the decomposition of extending lifting modules

장재훈,신종문 대한수학회 2009 대한수학회보 Vol.46 No.6

In 1984, Oshiro [11] has studied the decomposition of continuous lifting modules. He obtained the following: every continuous lifting module has an indecomposable decomposition. In this paper, we study extending lifting modules. We show that every extending lifting module has an indecomposable decomposition. This result is an expansion of Oshiro's result mentioned above. And we consider some application of this result. In 1984, Oshiro [11] has studied the decomposition of continuous lifting modules. He obtained the following: every continuous lifting module has an indecomposable decomposition. In this paper, we study extending lifting modules. We show that every extending lifting module has an indecomposable decomposition. This result is an expansion of Oshiro's result mentioned above. And we consider some application of this result.

Modules Which Are Lifting Relative To Module Classes

Kosan, Muhammet Tamer,Harmanci, Abdullah Department of Mathematics 2008 Kyungpook mathematical journal Vol.48 No.1

In this paper, we study a module which is lifting and supplemented relative to a module class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules and X-supplemented modules. Several properties of these modules are proved. We also obtain results for the case of specific classes of modules.

X-LIFTING MODULES OVER RIGHT PERFECT RINGS

장재훈 대한수학회 2008 대한수학회보 Vol.45 No.1

Keskin and Harmanci defined the family B(M,X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ HomR(M,X/Y ), Ker f/A ≪ M=A}. And Orhan and Keskin generalized projective modules via the class B(M,X). In this note we introduce X-local summands and X-hollow modules via the class B(M,X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ B(H,X), if HΘH has the internal exchange property, then H has a local endomorphism ring Keskin and Harmanci defined the family B(M,X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ HomR(M,X/Y ), Ker f/A ≪ M=A}. And Orhan and Keskin generalized projective modules via the class B(M,X). In this note we introduce X-local summands and X-hollow modules via the class B(M,X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ B(H,X), if HΘH has the internal exchange property, then H has a local endomorphism ring

Direct Sums of Strongly Lifting Modules

Shahabaddin Ebrahimi Atani,Mehdi Khoramdel,Saboura Dolati Pishhesari 경북대학교 자연과학대학 수학과 2020 Kyungpook mathematical journal Vol.60 No.4

For the recently defined notion of strongly lifting modules, it has been shown that a direct sum is not, in general, strongly lifting. In this paper we investigate the question: When are the direct sums of strongly lifting modules, also strongly lifting? We introduce the notion of a relatively strongly projective module and use it to show if M = M1+M2 is amply supplemented, then M is strongly lifting if and only if M1 and M2 are relatively strongly projective and strongly lifting. Also, we consider when an arbitrary direct sum of hollow (resp. local) modules is strongly lifting.

X-LIFTING MODULES OVER RIGHT PERFECT RINGS

( Jong Moom Shin ),( Chae Hoon Chang ) 한국수학교육학회 2014 純粹 및 應用數學 Vol.21 No.2

Keskin and Harmanci defined the family β(M,X) = {A ≤ MI ∃Y ≤ X, ∃f ∈ HomR(M,X/Y), Ker f/A ≪ M/A}. And Orhan and Keskin generalized projective modules via the class β(M,X). In this note we introduce X-local summands and X-hollow modules via the class β(M,X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module contains its radical, then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang``s result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ β(M,X), if H ㅁ H has the internal exchange property, then H has a local endomorphism ring.

X-LIFTING MODULES OVER RIGHT PERFECT RINGS

Chang, Chae-Hoon Korean Mathematical Society 2008 대한수학회보 Vol.45 No.1

Keskin and Harmanci defined the family B(M,X) = ${A{\leq}M|{\exists}Y{\leq}X,{\exists}f{\in}Hom_R(M,X/Y),\;Ker\;f/A{\ll}M/A}$. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class B(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang's result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with $K{\in}B$(H, X), if $H{\oplus}H$ has the internal exchange property, then H has a local endomorphism ring.

ON THE DECOMPOSITION OF EXTENDING LIFTING MODULES

Chang, Chae-Hoon,Shin, Jong-Moon Korean Mathematical Society 2009 대한수학회보 Vol.46 No.6

In 1984, Oshiro [11] has studied the decomposition of continuous lifting modules. He obtained the following: every continuous lifting module has an indecomposable decomposition. In this paper, we study extending lifting modules. We show that every extending lifting module has an indecomposable decomposition. This result is an expansion of Oshiro's result mentioned above. And we consider some application of this result.

Some Results on δ-Semiperfect Rings and δ-Supplemented Modules

ABDIOGLU, CIHAT,SAHINKAYA, SERAP Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.2

In [9], the author extends the definition of lifting and supplemented modules to ${\delta}$-lifting and ${\delta}$-supplemented by replacing "small submodule" with "${\delta}$-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of ${\delta}$-lifting and ${\delta}$-supplemented modules. Especially, we show that any finite direct sum of ${\delta}$-hollow modules is ${\delta}$-supplemented. On the other hand, the notion of amply ${\delta}$-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply ${\delta}$-supplemented and satisfies Descending Chain Condition (DCC) on ${\delta}$-supplemented modules and on ${\delta}$-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a ${\delta}$-semiperfect ring which satisfies DCC on ${\delta}$-small right ideals of R.

질소로 희석된 프로판 동축류 층류 제트 부상화염에서 열손실에 의한 자기진동에 대한 동축류 속도 효과

이원준(Won June Lee),윤성환(Sung Hwan Yoon),박정(Jeong Park),권오붕(Oh Boong Kwon),박종호(Jong ho Park),김태형(Tae hyung Kim) 한국연소학회 2012 한국연소학회지 Vol.17 No.1

Laminar lifted propane coflow-jet flames diluted with nitrogen were experimentally investigated to determine heat-loss-related self-excitation regimes in the flame stability map and elucidate the individual flame characteristics. There exists a critical lift-off height over which flame-stabilizing effect becomes minor, thereby causing a normal heat-loss-induced self-excitation with O(0.01 Hz). Air-coflowing can suppress the normal heat-loss-induced self-excitation through increase of a Peclet number; meanwhile it can enhance the normal heat-loss-induced self-excitation through reducing fuel concentration gradient and thereby decreasing the reaction rate of trailing diffusion flame. Below the critical lift-off height. the effect of flame stabilization is superior, leading to a coflow-modulated heat-loss-induced self-excitation with O(0.001 Hz). Over the critical lift-off height, the effect of reducing fuel concentration gradient is pronounced, so that the normal heat-loss-induced self-excitation is restored. A newly found prompt self-excitation, observed prior to a heat-loss-induced flame blowout, is discussed. Heat-loss-related self-excitations, obtained laminar lifted propane coflow-jet flames diluted with nitrogen, were characterized by the functional dependency of Strouhal number on related parameters. The critical lift-off height was also reasonably characterized by Peclet number and fuel mole fraction.