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20 February 2023
Volume 58 Issue 2

On the boundary factors of 2-maximal subgroups of finite groups

CHEN Xin-dan, XU Li, MIAO Long, LIU Wei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  1-5.  doi:10.6040/j.issn.1671-9352.0.2022.047

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Some 2-maximal subgroups of a finite group G are given, and the influence of the boundary factors of these 2-maximal subgroups on the structure of some classes of non-solvable groups is studied.

Pre-Lie-Yamaguti color algebra and color O-operator

TENG Wen, YOU Tai-jie

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  6-12.  doi:10.6040/j.issn.1671-9352.0.2022.010

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The definition of pre-Lie-Yamaguti color algebra is introduced. The color O -operator of Lie-Yamaguti color algebra about representation space is given, and its relationship with pre-Lie-Yamaguti color algebra is discussed. It is proved that a pre-Lie-Yamaguti color algebra can induce a Lie-Yamaguti color algebra and a representation on itself.

The number of endomorphisms and automorphisms of a class of non-abelian groups

ZHANG Wei, GUO Ji-dong, ZHANG Liang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  13-19.  doi:10.6040/j.issn.1671-9352.0.2022.058

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Based on the group structure and the order of elements of a class of non-abelian groups in group theory, the number of endomorphisms and automorphisms of a class of non-abelian groups of order 2qpⁿ whose Sylow p-subgroups is cyclic is calculated, where q<p and both are odd primes. Moreover, it is proved that the number of endomorphisms of such groups satisfies the conjecture of T. Asai and T. Yoshida in this case.

Congruences on type B semigroups

LI Chun-hua, MENG Ling-xiang, FANG Jie-ying

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  20-27.  doi:10.6040/j.issn.1671-9352.0.2021.795

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The main aim of this paper is to study admissible congruences on a type B semigroup. Some properties of kernels of admissible congruences on a type B semigroup are given, and a characterization of the minimum admissible congruence on a type B semigroup is obtained. In particular, some characterizations of admissible congruences with E property on a type B semigroup are given, and some results are obtained.

The quasi-Baer property of the *-skew polynomial rings

WANG Yao, QIN Lan-lan, REN Yan-li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  28-32.  doi:10.6040/j.issn.1671-9352.0.2021.809

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The *-principally quasi-Bear property and quasi-Bear-*-property of the *-skew polynomial rings are investigated. It is proved that(1)Let R be a *-right principally quasi-Baer ring, if re=0 implies that re^*=0 for any e∈S^*_l(R) and r∈R, then R［x;*］ is a *-right principally quasi-Baer ring;(2)Let * be a true involution on R and R be *-reversible, then R［x;*］ is a quasi-Baer *-ring if and only if R is a quasi-Baer *-ring.

Gröbner-Shirshov basis of modified Ringel-Hall algebra of type B₂

HU De-sheng, Abdukadir OBUL

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  33-43.  doi:10.6040/j.issn.1671-9352.0.2022.200

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Skew commutator relations between the isoclasses of indecomposable complexes in the modified Ringel-Hall algebra of type B₂ is computed, it is proved that the set of these skew commutator relations forms a minimal Gröbner-Shirshov basis for the modified Ringel-Hall algebra of type B₂. As an application, a PBW-type basis of the modified Ringel-Hall algebra of type B₂ is obtained.

Maschke theorem for the Grothendieck algebra of dihedral group

CAO Liu-feng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  44-50.  doi:10.6040/j.issn.1671-9352.0.2021.725

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Let F be an algebraically closed field with characteristic 0, and n=2N+1 be any odd number, we calculating the Casimir number of the Grothendieck ring r(FD_n) of the dihedral group D_n equals 2n2 explicitly. Furthermore, we give the Maschke theorem for the corresponding Grothendieck algebra of dihedral group.

n-Frobenius pairs

CHEN Wen-jing, LI Ling

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  51-57.  doi:10.6040/j.issn.1671-9352.0.2022.054

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The definitions of left(right)n-Frobenius pairs and special left(right)n-Frobenius pairs are introduced. Their properties are studied. The relationship between left(right)n-Frobenius pairs and right(left)n-cotorsion pairs is discussed, and some equivalent characterizations of special left(right)n-Frobenius pairs are given. Finally, some applications of special left(right)n-Frobenius pairs and equivalent characterizations of the left perfect QF ring are given.

Strongly Ding projective modules over Morita rings

TAN Jin, DI Zhen-xing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  58-62.  doi:10.6040/j.issn.1671-9352.0.2021.669

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This paper constructs a class of strongly Ding projective modules on Morita rings with zero bimodule homomorphisms.

(n,m)-Strongly projectively coresolved Gorenstein flat modules

ZHONG Kui-chen, ZHANG Cui-ping, QIN Jun-xia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  63-71.  doi:10.6040/j.issn.1671-9352.0.2021.660

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The concept of an (n,m)-strongly projectively coresolved Gorenstein flat module(i.e.(n,m)-strongly PGF module)is introduced, and some basic properties of it are given. It is proved that if M is an (n,m)-strongly PGF module, then(1)PGF dimension of M PGFd(M)≤m;(2)when 1≤i≤m, the ith syzygy of M is(n,m-i)-strongly PGF module; when i≥m, the ith syzygy of M is(n,0)-strongly PGF module. Secondly, it is proved that the dth syzygy of M is(1,m)-strongly PGF module, then PGF d(M)=k≤d+m, and M is (1,k)-strongly PGF module.

A note on DG-Gorenstein AC-injective complexes

WANG Xin, LU Bo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  72-78.  doi:10.6040/j.issn.1671-9352.0.2022.201

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The notion of DG-Gorenstein AC-injective complexes is introduced and studied. It is proven that over any ring a complex G is DG-Gorenstein AC-injective if and only if G is exact with Z_n(G)Gorenstein AC-injective in R-Mod for any integer n and any homomorphism f:A→G is null homotopic whenever A is a DG-FP_∞-injective complex.

On (n,d)-phantom and (n,d)-Ext-phantom morphisms

ZHANG Chun-xia, TANG Qiang-ling

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  79-87.  doi:10.6040/j.issn.1671-9352.0.2022.094

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Let n,d be nonnegative integers and n≥1, we introduce the concepts of(n,d)-phantom and(n,d)-Ext-phantom morphisms, and give their some properties. As applications, we obtain some new characterizations of the FPn-flat and FPn-injective dimensions of modules.

Finitely Gorenstein copresented comodules over triangular matrix coalgebras

GUO Chun-na, YAO Hai-lou

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  88-92.  doi:10.6040/j.issn.1671-9352.0.2021.683

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The specific form of the finitely Gorenstein copresented comodule over a triangular matrix coalgebra Γ=(T _TM_U0 U)is given by the equivalence theorem of categories and the exact functors between categories, and a relationship between the finitely Gorenstein copresented dimensions of the triangular matrix coalgebra Γ and coalgebras T and U is obtained as follows Max{G.cp.dimT,G.cp.dimU}≤G.cp.dimΓ≤G.cp.diimT+G.cp.dimU+1.

Modified Rota-Baxter paired modules and modified Rota-Baxter paired comodules

FANG Ying, ZHENG Hui-hui, ZHANG Liang-yun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  93-104.  doi:10.6040/j.issn.1671-9352.0.2021.805

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The concepts of modified Rota-Baxter paired module and modified Rota-Baxter paired comodule are introduced. Then modified Rota-Baxter paired module and modified Rota-Baxter paired comodule are constructed by Hopf algebras respectively. Finally, the concept of modified Rota-Baxter paired Hopf module is introduced, and its structure theorem is given.

(n,d)-Injective modules over triangular matirx rings

DAIQING Ang-mao, LU Bo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(2):  105-110.  doi:10.6040/j.issn.1671-9352.0.2022.006

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Let A, B be rings, U a (B,A)-bimodule, n, d non-negative integers, T=(A 0U B)a formal triangular matrix ring, firstly, it is shown that M=(M1M2)_φ^M is an n-presented left T-module if and only if M₁ is an n-presented left A-module, Coker φ^M is an n-presented left B-module and φ^M:U_AM1 → M2 is a monomorphism. Secondly, it is shown that M₁ is an (n,d)-injective left A-module and M₂ is an(n,d)-injective left B-module whenever M=(M1M2)_φ^M is an(n,d)-injective left T-module.