JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

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20 July 2020
Volume 55 Issue 8

On torsion free class and cover class in category of comodules

LI Yuan, YAO Hai-lou

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 1-5. doi:10.6040/j.issn.1671-9352.0.2019.869

Abstract ( 1301 )

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For a coalgebra, the concept of(pre)covers for comodules is first introduced and some properties about them are given. Then, the concepts of maximal tilting comodules and cover comodules are given, and the existence of a bijection between the tilting torsion free classes and the maximal tilting comodules is proved. Finally, when the tilting torsion free class is a cover class, whose unique representation by cover comodules over a coalgebra is obtained.

Lie H-pseudoalgebras over triangular Hopf algebras

ZHANG Yong-feng, YANG Shi-lin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 6-12. doi:10.6040/j.issn.1671-9352.0.2020.025

Abstract ( 910 )

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The purpose of this paper is to determine Lie H₄-pseudoalgebra structures on a free left H₄-module L of rank one, where H₄ is the Sweedlers Hopf algebra.

(Strongly)J-*-tripotent-clean rings

ZHENG Ji-wen, CHENG Zhi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 13-19. doi:10.6040/j.issn.1671-9352.0.2019.436

Abstract ( 917 )

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The concept of(strongly)J-*-tripotent-clean ring is introduced. An example is given to illustrate that strongly J-*-tripotent-clean ring is a proper subclass of strongly clean ring. Some equivalent definitions of strongly J-*-tripotent-clean ring are given. As an application, some corresponding properties for transformations of rings are also given.

Strongly g(x)-J^#-clean rings

LIU Song-song, WU Jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 20-27. doi:10.6040/j.issn.1671-9352.0.2019.440

Abstract ( 969 )

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The concept of strongly g(x)-J^#-clean rings are introduced. Some properties of strongly g(x)-J^#-cleanrings are obtained. Moreover, equivalent characterizations of strongly J^#-cleanrings are given by means of strongly g(x)-J^#-clean rings.

Gröbner-Shirshov basis of irreducible modules over quantum group of type B2

HE Yu-xing, Abdukadir Obul

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 28-37. doi:10.6040/j.issn.1671-9352.0.2020.024

Abstract ( 994 )

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By using the known Gröbner-Shirshov basis of quantum group U_q(B2) and the double free module method, we give a Gröbner-Shirshov pair of finite dimensional irreducible U_q(B2)-module V_q(λ), first, then by specializing a suitable version of U_q(B2) at q=1, we get a Gröbner-Shirshov basis of U(B2) and a Gröbner-Shirshov pair of finite dimensional irreducible U(B2)-module V(λ).

Abelian extensions of δ-Lie color algebras

MA Li-li, LI Qiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 38-42. doi:10.6040/j.issn.1671-9352.0.2019.877

Abstract ( 920 )

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We construct δ-Lie color algebra T⊕V by representations and 2-cocycle of δ-Lie color algebra T. Then we prove that equivalent abelian extensions of δ-Lie color algebras give the same representation. Finally, we obtain a 2-cocycle using representations and its abelian extension.

Limit categories of complete categories and η-extensions

HUANG Ju, FAN Xin-xiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 43-47. doi:10.6040/j.issn.1671-9352.0.2019.736

Abstract ( 1554 )

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The relationships between the limit categories and the η-extension of complete categories are researched. Then the equivalence between the η-extension of a limit category and the limit category of an η-extension as a complete category is shown.

K_i-group of t-structure of triangulated category

ZHENG Min, CHEN Qing-hua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 48-53. doi:10.6040/j.issn.1671-9352.0.2019.912

Abstract ( 1073 )

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Based on the properties of t-structure of a triangulated category, the paper discusses the relationship between the K_i-groups of t- structure of the triangulated category and K_i-groups of the triangulated category. Moreover, if t-structure is stable, it is shown that K_i- group of triangulated category is isomorphic to the direct sum of K_i- groups of t- structure.

Some characterizations of a class of special morphisms of modules

LAN Kai-yang, YANG Ting-ting

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 54-58. doi:10.6040/j.issn.1671-9352.0.2019.692

Abstract ( 1012 )

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Let n be a positive integer. First, some equivalent characterizations of Torn-monomorphism and Extn-epimorphism are studied. Secondly, some sufficient conditions for a morphism to be a Torn-epimorphism or an Extn-monomorphism are given.

f-injective modules with respect to semidualizing modules

LAN Kai-yang, LU Bo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 59-64. doi:10.6040/j.issn.1671-9352.0.2019.648

Abstract ( 1254 )

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Let R be a commutative ring and C a semidualizing R-module. The f-injective modules with respect to a semidualizing R-module C is defined and studied, and it is proved that a homomorphism F→M of R-modules is an injective(f-injective)precover of M if and only if HomR(C,F)→HomR(C,M) is a C-injective(C-f-injective)precover.

PBW-deformations of type B₂ quantum group and their symmetry property

REN Xiao-qian, XU Yong-jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 65-74. doi:10.6040/j.issn.1671-9352.0.2019.540

Abstract ( 1332 )

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The negative parts of quantum groups are very important connected graded algebras appeared in the quantum group theory. Their PBW-deformations are called the PBW-deformations of quantum groups. Except the A2 case, the defining relations of these algebras are homogeneous quantum Serre relations with mixed degrees. In particular, the defining relations of the negative part of type B₂ quantum group are respectively the homogeneous quantum Serre relations with degree 3 and 4. In this paper, in the framework of PBW-deformations of connected graded algebras, we explicitly characterize all the PBW-deformations of type B₂ quantum group and investigate their symmetry property, i.e., all the symmetric PBW-deformations of type B₂ quantum group under four kinds of involution anti-automorphisms are obtained.

Simple Utumi modules

WU De-jun, LU Rui-tao, WANG Yong-duo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 75-79. doi:10.6040/j.issn.1671-9352.0.2019.538

Abstract ( 957 )

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Simple Utumi modules and singular simple Utumi modules are introduced, and their fundamental properties are studied. It is proved that a ring R is a right V-ring if and only if each right R-module is a simple Utumi module, a ring R is a right GV-ring if and only if each right R-module is a singular simple Utumi module and a ring R is a semisimple Artinian ring if and only if each simple Utumi right R-module is an injective module.

Number of homomorphisms between central-dihedral group and dihedral group

XIE Wei, GUO Ji-dong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 80-86. doi:10.6040/j.issn.1671-9352.0.2019.839

Abstract ( 1456 )

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Based on the group structure and element properties of central-dihedral group and dihedral group in group theory, the number of homomorphisms between central-dihedral group and dihedral group is calculated by using the related theories of algebra and number theory. As an application, the conjecture of T.Asai & T.Yoshida is proved to be valid for such groups.

Structure of weakly type σ semigroups

GONG Chun-mei, GAO Wen, YUAN Ying

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 87-91. doi:10.6040/j.issn.1671-9352.0.2019.240

Abstract ( 1031 )

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The definition of weakly type σ semigroups is given and the quasispinded product structure of weakly type σ semigroups is established. It is proved that a semigroup S is a weakly typeσ semigroup if and only ifS is a quasispinded product of a semiadequate semigroup T and a left regular band I.

A classs of finite p-groups with few normal subgroups

WANG Jun-xin, BAI Peng-fei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 92-97. doi:10.6040/j.issn.1671-9352.0.2019.438

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The N1-p-groups are completely classified. The so-called N₁-group is a group G that has only one normal subgroup which neither contains G' nor is contained in Z(G).

(*)-Good congruences on superabundant semigroups

GONG Chun-mei, WANG Hui, WU Dan-dan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 98-101. doi:10.6040/j.issn.1671-9352.0.2019.508

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The characterizations of(*)-good congruence on superabundant semigroups are studied by means of(*)-Greens relation and the properties of superabundant semigroups. The sufficient and necessary condition that any two elements on a superabundant semigroup S are in the same(*)-good congruence class is given and the description of any congruence contained in D ^* on S is obtained.

Global structure of the positive solution for a class of fourth-order boundary value problems with first derivative

ZHANG Ya-li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(8): 102-110. doi:10.6040/j.issn.1671-9352.0.2019.725

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This paper considers the global structure of positive solutions for a fourth-order boundary value problem with first derivative{u⁽⁴⁾(t)=rf(t,u(t),u'(t)), t∈(0,1),u(0)=u'(0)=u″(1)=u(1)=0,where r is a positive parameter, f:［0,1］×［0,∞)×［0,∞)→［0,∞)is continuous, and f(t,0,0)=0. When the parameter r changes in a certain range, the global structure of positive solutions of the problem are obtained by using the Rabinowitz global bifurcation theorems. The conclusions in this paper generalize and improve the related results.