DEPTH FOR TRIANGULATED CATEGORIES

Liu, Yanping,Liu, Zhongkui,Yang, Xiaoyan Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2

Recently a construction of local cohomology functors for compactly generated triangulated categories admitting small coproducts is introduced and studied by Benson, Iyengar, Krause, Asadollahi and their coauthors. Following their idea, we introduce the depth of objects in such triangulated categories and get that when (R, m) is a graded-commutative Noetherian local ring, the depth of every cohomologically bounded and cohomologically finite object is not larger than its dimension.

Direct Sums of Extending Modules

Liu Zhongkui 경북대학교 자연과학대학 수학과 2003 Kyungpook mathematical journal Vol.43 No.1

Let M and N be left R-modules. N is called M -ep-injective if N is essentially M -injective and pseudoly M -injective. An example is given to show that M -ep-injectivityis a non-trivial generalization of M -injectivity. We show that any direct sum of relatively ep-injective extending left R-modules of finite length is special extending. It is also proved that the direct sum M = i2 I M i of left R-modules M i (i 2 I,jIj 2) is extending if and only if there exist i 6= j in I such that every closed submodule K of M with K \ M i e Kor K \ M j e K or K \ M i = K \ M j = 0 is a direct summand.

ON RELATIVE COHEN-MACAULAY MODULES

Zhongkui Liu,Pengju Ma,Xiaoyan Yang Korean Mathematical Society 2023 대한수학회지 Vol.60 No.3

Let a be an ideal of 𝔞 commutative noetherian ring R. We give some descriptions of the 𝔞-depth of 𝔞-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaulayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.

Depth for triangulated categories

Yanping Liu,Zhongkui Liu,Xiaoyan Yang 대한수학회 2016 대한수학회보 Vol.53 No.2

Recently a construction of local cohomology functors for compactly generated triangulated categories admitting small coproducts is introduced and studied by Benson, Iyengar, Krause, Asadollahi and their coauthors. Following their idea, we introduce the depth of objects in such triangulated categories and get that when $(R,\mathfrak{m})$ is a graded-commutative Noetherian local ring, the depth of every cohomologically bounded and cohomologically finite object is not larger than its dimension.

On some properties of Malcev-Neumann modules

Renyu Zhao,Zhongkui Liu 대한수학회 2008 대한수학회보 Vol.45 No.3

Let M be a right R-module, G an ordered group and σ 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 MR is a reduced σ-compatible module, then the Malcev-Neumann module M * ((G)) over a PS-module is also a PS-module; (2) If MR is a faithful σ-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. Let M be a right R-module, G an ordered group and σ 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 MR is a reduced σ-compatible module, then the Malcev-Neumann module M * ((G)) over a PS-module is also a PS-module; (2) If MR is a faithful σ-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.

ON SOME PROPERTIES OF MALCEV-NEUMANN MODULES

Zhao, Renyu,Liu, Zhongkui Korean Mathematical Society 2008 대한수학회보 Vol.45 No.3

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.

Ding projective modules with respect to a semidualizing module

Chunxia Zhang,Limin Wang,Zhongkui Liu 대한수학회 2014 대한수학회보 Vol.51 No.2

In this paper, we introduce and discuss the notion of DC- projective modules over commutative rings, where C is a semidualizing module. This extends Gillespie and Ding, Mao’s notion of Ding projective modules. The properties of DC-projective dimensions are also given.

Extensions of Strongly α-semicommutative Rings

Ayoub, Elshokry,Ali, Eltiyeb,Liu, ZhongKui Department of Mathematics 2018 Kyungpook mathematical journal Vol.58 No.2

This paper is devoted to the study of strongly ${\alpha}-semicommutative$ rings, a generalization of strongly semicommutative and ${\alpha}-rigid$ rings. Although the n-by-n upper triangular matrix ring over any ring with identity is not strongly ${\bar{\alpha}}-semicommutative$ for $n{\geq}2$, we show that a special subring of the upper triangular matrix ring over a reduced ring is strongly ${\bar{\alpha}}-semicommutative$ under some additional conditions. Moreover, it is shown that if R is strongly ${\alpha}-semicommutative$ with ${\alpha}(1)=1$ and S is a domain, then the Dorroh extension D of R by S is strongly ${\bar{\alpha}}-semicommutative$.

DING PROJECTIVE MODULES WITH RESPECT TO A SEMIDUALIZING MODULE

Zhang, Chunxia,Wang, Limin,Liu, Zhongkui Korean Mathematical Society 2014 대한수학회보 Vol.51 No.2

In this paper, we introduce and discuss the notion of $D_C$-projective modules over commutative rings, where C is a semidualizing module. This extends Gillespie and Ding, Mao's notion of Ding projective modules. The properties of $D_C$-projective dimensions are also given.

Experimental Study on Soil Deformation during Sampler Penetration

Chuanyang Liang,Jian Liu,Yuedong Wu,Zhongkui Chen,Ruping Luo,Jin Zheng,Yike Meng 대한토목학회 2022 KSCE JOURNAL OF CIVIL ENGINEERING Vol.26 No.3

Disturbance at the stage of penetration during soil sampling induces the deformation of samples, causing unreliable results on the determination of soil properties. In order to minimize the disturbance on soil during the penetration process, the displacement field around the tube sampling and influence factors on disturbance are investigated widely. However, there is a lack of effective observation on the development of soil movement at axis and on cross sections with the increase in the penetration depth, which should be studied individually at a certain penetration depth. This paper presents the effect of different sampler penetration depths on the development of soil at central axis and on cross sections through a series of 1-g physical model tests using the particle image velocimetry (PIV) technique. Results show that with the increase in the penetration depth, the correlation between the maximum compression deformation at central axis and the depth corresponding location may be approximated a hyperbola. Moreover, the surface of clay presents the downward compression deformation while that of sand presents the upward deformation. Furthermore, the disturbance influence depth of soil at central axis increases approximately linearly with the increase in the penetration depth. For the sand, the correlation between sensitivity on cross sections and depth presents approximately a line while that of clay presents a hyperbola. These deformation correlations provide useful references for detail predicting the overall deformation curve at a certain penetration depth, which helps to further develop and validate analytical or numerical solutions for this problem.