JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Some studies on condition(IE)

QIAO Husheng, ZHANG Xueqian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 1-5. doi:10.6040/j.issn.1671-9352.0.2023.499

Abstract ( 126 )

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Let S be a monoid, and K be a right ideal of S. A generalized condition(IE)of condition(E)is introduced. The relationship between condition(IE)and condition(E)and other flat properties is studied. The coproduct, amalgamated coproduct and direct product of S-acts satisfying condition(IE)are discussed. In addition, the necessary and sufficient conditions for the cyclic acts S/ρ, diagonal acts D(S) and Rees quotient acts S/K to satisfy condition(IE)are given.

Rank of the 2-singular transformation semigroup T_n(2)

GAO Ronghai, XU Bo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 6-10. doi:10.6040/j.issn.1671-9352.0.2023.545

Abstract ( 147 )

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Let n≥4, X_n={1,2,…,n}. Singn is singular transformation semigroup on X_n. for α∈Singn, α is called 2-singular transformation if it exists x∈X_n\{1} that satisfy xα=1α or exists x∈X_n\{2} that satisfy xα=2α. Let T_n(2) be consisting of all 2-singular transformations in Singn. A minimum generating set is determined through the construction method, and it is proved that the rank of T_n(2) is 2n-3.

Influence of traces of some subgroups on solvable groups

ZHANG Jia, HE Jinlü, XIANG Yanhui, WU Jinlian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 11-15. doi:10.6040/j.issn.1671-9352.0.2023.397

Abstract ( 109 )

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The influence of the supersolvable property of trace of primitive groups(which cannot be written as a proper intersection of subgroups)on the solvability of groups is analyzed. By dividing all maximal subgroups into two classes F_p (maximal subgroups whose indices are prime number), F_c (maximal subgroups whose indices are composite number), the connection between the nilpotency of the trace of every maximal subgroup in F_c and the structure of solvable groups is studied. By extending maximal subgroups to second maximal subgroups, three sufficient and necessary conditions on solvable groups are obtained.

Characterizations of EP elements and bi-EP elements

LI Tingting, GAO Yuefeng, KE Yuanyuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 16-19. doi:10.6040/j.issn.1671-9352.0.2023.312

Abstract ( 116 )

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Let R be a ring and a∈R. When a∈R^#∩R†, we give some characterizations such that a is an EP element or a bi-EP element by the Moore-Penrose inverse and group inverse of elements in {a,a^#,(a†)^*,a†,a^*,(a^#)^*}.

A new characterization of A₅ and PSL(2,7)

KUANG Meiqun, LU Jiakuan, LI Yu, ZHANH Boru

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 20-26. doi:10.6040/j.issn.1671-9352.0.2023.258

Abstract ( 142 )

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Let G be a finite group, and let m(G)=∑_g∈G1/(o(g)), where o(g) denotes the order of g∈G, and h(G) is the maximal order of the elements in G. We give a new characterization of A5 and PSL(2,7), that is, G≅A5 if and only if m(G)=m(A5) and h(G)=h(A5). G≅ PSL(2,7)if and only if m(G)=m(PSL(2,7))and h(G)=h(PSL(2,7)).

On the rank of the semigroup P_{O_m}(n)

GOU Changsheng, RUAN Limin, ZHAO Ping

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 27-31. doi:10.6040/j.issn.1671-9352.0.2024.130

Abstract ( 134 )

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Let P_n be a partial transformation semigroup on the finite set X_n={1,2,…,n}. Let 1≤m≤n, a subsemigroup of P_n be defined byP_{O_m}(n)={α∈P_n:X_m⊆dom(α), α|_{X_m}∈O_m},In the paper, the rank of the subsemigroup P_{O_m}(n)is studied, where 1≤m≤n-1. For n≥3, it is proven thatrank P_{O_m}(n)={2+2m, n-m=1,3+2m, n-m=2,4+2m, n-m≥3.

The influence of subnormal subgroups on the n-solvability of groups

BAI Yiman, HAI Jinke

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 32-36. doi:10.6040/j.issn.1671-9352.0.2025.014

Abstract ( 91 )

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Let n be an integer(positive or negative or 0). In this paper, we prove that the group 〈H,K〉 generated by two finite n-soluble groups H and K is still a finite n-soluble group if one of the subgroups is subnormal. Moreover, it is proved that if all non-n-nilpotent proper subgroups of a finite group G are subnormal and n-soluble, then G is n-soluble.

X -Gorenstein cohomological dimension for groups

LEI Yiming, LIANG Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 37-41. doi:10.6040/j.issn.1671-9352.0.2023.428

Abstract ( 120 )

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The X -Gorenstein cohomology dimension of groups was introduced and the finiteness of the X -Gorenstein cohomology dimension of groups was characterized.

Small finitistic dimension of Abel category in a recollement

WANG Xi, YAO Hailou

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 42-47. doi:10.6040/j.issn.1671-9352.0.2024.204

Abstract ( 104 )

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Let A be an Abel category. The concept of FT-injective objects and the FT-injective dimension are introduced in A, and the basic properties of FT-injective objects and the FT-injective dimension are discussed. Additionally, using the FT-injective dimension, the small finitistic dimension of A, denoted as f PDA is characterized. The relationship among the small finitistic dimensions of three different Abel categories in a recollement is explored.

FI-injective N-complexes

HAN Xia, LU Bo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 48-53. doi:10.6040/j.issn.1671-9352.0.2023.498

Abstract ( 130 )

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Let n be an integer. The notion of FI-injective N-complexes is introduced and studied. It is proven that an N-complex I is FI-injective over a coherent ring if and only if each I_n is an FI-injective module and HomR(G,I) is N-exact whenever G is an FP-injective N-complex. The characterization of FI-injective N-complexes is given, and the relationship between it and injective N-complexes is considered.

Gorenstein homological objects in trivial extensions

QIN Xiaoyu, LIANG Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 54-58. doi:10.6040/j.issn.1671-9352.0.2023.470

Abstract ( 106 )

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By introducing the(co)compatibility of functors, we give the characterizations of Gorenstein homological objects in trivial extensions of abelian categories.

The additively idempotent semiring variety V(S(a2b))

WU Yanan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 59-64. doi:10.6040/j.issn.1671-9352.0.2023.246

Abstract ( 117 )

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The variety V(S(a2b)) generated by S(a2b) is studied, and the lattice L(V(S(a²b)))of subvarieties of V(S(a2b)) is depicted. Also, each member of this lattice is finitely based and V(S(a2b)) is hereditarily finitely based are proved.

Full congruences of a semiring variety

TAO Binghui, SHAO Yong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 65-69. doi:10.6040/j.issn.1671-9352.0.2023.429

Abstract ( 104 )

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The definitions of full equivalence relations and full congruences of a semiring variety are introduecd. A method of generating a full congruence with an arbitrary full equivalence relation is presented. Several varieties of semirings defined by full congruences are characterized and described. Some connections between these semring varieties are given and discussed. The results obtained generalize the relevant conclusions.

Quasi J-*-clean rings

LI Xiaoxuan, YIN Xiaobin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 70-78. doi:10.6040/j.issn.1671-9352.0.2023.553

Abstract ( 118 )

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A ring is called(strongly)quasi J-*-clean ring if each of its elements is a sum of a quasi-projection and an element of the Jacobson radical(that commute). The basic properties of this ring and some relations with other rings are studied. It is proved that R is a strongly quasi J-*-clean ring if and only if R is a strongly *-clean ring and R/J(R) is a strongly quasi J-*-clean ring if and only if R/J(R) is a strongly quasi J-*-clean ring, every projection of R is central and every projection of R/J(R) is lifted to a projection of R.

Gorenstein flat modules and dimensions with respect to duality pairs over formal triangular matrix rings

LIU Ling, CHEN Wenjing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 79-86. doi:10.6040/j.issn.1671-9352.0.2023.479

Abstract ( 129 )

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Let T=(A 0U B) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. G F_{(F(R),Y )} denotes the class of all Gorenstein flat left R-modules with respect to a complete duality pair(X,Y ). Assume that(C₁,C₂)and(D₁,D₂)are complete duality pairs over the ring A and the ring B respectively, and(U^C₁_D₁,U_C₂,D₂)is a complete duality pair over the ring T induced by(C₁,C₂)and(D₁,D₂). It is proven that, if U_A has finite flat dimension and G F_{(F(T),U_C₂,D₂)} is closed under extensions, then a left T-module (M1M2)_φ^M∈G F_{(F(T),U_C₂,D₂)} if and only if M1∈G F_(F(A),C₂), M₂/Im(φ^M)∈G F_(F(B),D₂), and φ^M:U⊗_AM1→M2 is a monomorphism. Furthermore, an estimate of Gorenstein flat dimension with respect to the complete duality pair(U^C₁_D₁,U_C₂,D₂)of a left T-module is given.

Generalized inverses in the representation ring of symmetric group S3

CAO Liufeng, WEI Junchao, HU Jiawei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 87-94. doi:10.6040/j.issn.1671-9352.0.2023.486

Abstract ( 208 )

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By using Frobenius-Perron dimension, all regular elements in the representation ring r(S3) of the symmetric group S3 of order 6 are determined, and all invertible elements, MP-inverses, group invertible elements, EP and SEP elements in r(S3) are described.

Strongly Gorenstein projective modules over Morita rings

XIA Xu, CHEN Wenjing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 95-100. doi:10.6040/j.issn.1671-9352.0.2023.476

Abstract ( 93 )

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Let Λ_ψ=(A NM B)₍0,ψ) be a Morita ring with one bimodule homomorphism zero, where A and B are noetherian rings, N is an A-B-bimodule, M is a B-A-bimodule, and ψ:N_BM→A is a homomorphism of A-A-bimodules. We give the sufficient condition that a Λ_ψ-module is a strongly Gorenstein projective module.

Subvarieties based on the semiring Greens relation in COS^·_n

WANG Aifa, HU Yu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 101-108. doi:10.6040/j.issn.1671-9352.0.2024.038

Abstract ( 104 )

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The Green's relations(·overL)∨D+,(·overL)∨L+,(·overL)∨R+, L+∨(·overD)of semirings are studied, and equivalent characterizations for these relations within semirings are provided. The necessary and sufficient conditions for these relations to constitute congruences are obtained. It is proven that the semiring classes determined by the aforementioned congruence relation are subvarieties. The Mal'cev product decomposition of these subvarieties is given.

Strongly Gorenstein projective modules over trivial ring extensions

HE Jianyuan, JIN Yuanhui, WANG Zhanping

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 109-114. doi:10.6040/j.issn.1671-9352.0.2023.484

Abstract ( 159 )

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Let RM be a trivial ring extension, where R is a ring and M is an R-R-bimodule. We give sufficient and necessary conditions such that (X,α) is a strongly Gorenstein projective left RM-module.

A syzygy-infinite self-injective algebras

ZHOU Jianguo, LIU Yuzhe, ZHANG Chao, ZHANG Yafeng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 115-121. doi:10.6040/j.issn.1671-9352.0.2023.391

Abstract ( 195 )

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Let Λ be a finite dimensional k-algebra with an algebraically closed field k. It is well-known that if Λ is syzygy-finite then, by using that Λ is an n-Igusa-Todorov algebra, its finitistic dimension is finite. This paper shows that the inverse of the above proposition is false by the enveloping algebra of some Nakayama algebras, that is, there exists an algebra with finite finitistic dimension such that it is a syzygy-infinite algebra.

The biderivation of Lie color algebras

LI Xinyang, SUN Bing, ZHOU Xin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 122-129. doi:10.6040/j.issn.1671-9352.0.2023.366

Abstract ( 118 )

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In this paper, the biderivations of Lie color algebras are defined, and the internal relations among biderivations, commuting maps and centroids of Lie color algebras satisfying certain conditions are established.

The idempotents of the quantum double of 16-dimensional Taft algebra

YIN Zetao, HU Chengchao, CHEN Huixiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 130-133. doi:10.6040/j.issn.1671-9352.0.2023.324

Abstract ( 97 )

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The idempotents of the quantum double of 16-dimensional Taft algebra are investigated. According to the known conclusion of the quantum double of Taft algebra, a complete set of primitive orthogonal idempotents of the quantum double of 16-dimensional Taft algebra are constructed.

Lie triple derivations on a generalized matrix algebra

ZHUANG Jinhong, CHEN Yanping, TAN Yijia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 134-147. doi:10.6040/j.issn.1671-9352.0.2023.381

Abstract ( 152 )

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The structure of Lie triple derivations at zero point on a generalized matrix algebra is studied,and an equivalent condition for a Lie triple derivation at zero point to be expressed as the sum of a derivation, a singular Jordan derivation and a linear map from the generalized matrix algebra to its center is obtained. Partial results obtained in the paper generalize the corresponding results for triangular algebras.

Characterizations of Gorenstein B-injective complexes

LI Qihui, YANG Gang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 148-152. doi:10.6040/j.issn.1671-9352.0.2023.549

Abstract ( 109 )

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( ~)/B denotes the class of all B -complexes, and B denotes a class of modules which contains all injective R-modules. It is proved that N is a Gorenstein( ~)/B -injective complex if and only if N_i is a Gorenstein B -injective module for each i∈Z and Hom(B,N)is exact for any B -complex B.

The ce-topology and cρ-topology on consistently continuous posets

GUO Zhilian, YANG Hailong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2025, 60(11): 153-158. doi:10.6040/j.issn.1671-9352.0.2024.085

Abstract ( 86 )

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The definitions of ce-topology and cρ-topology on consistently directedly complete posets are introduced. Some properties of them on consistently continuous posets are given. Furthermore, it is obtained that the basis on a consistently continuous partial order set is topologically dense.

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