Algebra Colloquium

The action of PSL(2,ℤ) on PL(F_5ⁿ) yields PSL(2,5). In this paper, we use the adjacency matrices of the abstract group PSL(2,5), eventually of fullerene C₆₀, to draw the coset diagram and investigate various properties of the diagram with reference to the intransitivity of the action and the number of orbits.

In this paper we mainly study sporadic groups in terms of p-local rank, especially, the 2-local ranks of some sporadic groups are given.

Eick and Leedham-Green introduced infinite sequences of finite p-groups of fixed coclass. They showed that the groups in such an infinite sequence can be defined by a single parametrised presentation. Our aim is to construct short or even efficient parametrised presentations for such infinite sequences of finite 2-groups.

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H₁⊕ ⋯ ⊕ H_m homogeneous in G-grading with graded simple summands H₁,…,H_m. All SuppH_i (i=1,…,m) are commutative subsets of G.

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified.

Let R be a prime ring of characteristic different from 2, L a noncentral Lie ideal of R, H and G two nonzero generalized derivations of R. Suppose u^s(H(u)u-uG(u)) u^t=0 for all u ∈ L, where s, t ≥ 0 are fixed integers. Then either (i) there exists p ∈ U such that H(x)=xp for all x ∈ R and G(x)=px for all x ∈ R unless R satisfies S₄, the standard identity in four variables; or (ii) R satisfies S₄ and there exist p, q ∈ U such that H(x)=px+xq for all x ∈ R and G(x)=qx+xp for all x ∈ R.

An R-module M is called Ding projective if there exists an exact sequence ⋯ → P₁→ P₀→ P⁰→ P¹→ ⋯ of projective R-modules with M=Ker(P⁰→ P¹) such that Hom(-,F) leaves the sequence exact whenever F is a flat R-module. In this paper, we develop some basic properties of such modules. Also, properties of Ding injective modules are discussed.

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f(x₁,…,x_n) a multilinear polynomial over C, a ∈ R, and I a nonzero right ideal of R. Suppose that a[g(f(r₁,…,r_n)), f(r₁,…,r_n)]=0 for all r_i∈ I and aI ≠ 0. Then either g(x)=a₁x with (a₁-γ)I=0 for some a₁∈ U and γ ∈ C, or there exists an idempotent element e ∈soc(RC) such that IC=eRC and one of the following holds: (i) f(x₁,…,x_n) is central-valued in eRe; (ii) g(x)=bx+xc, where b, c ∈ U with (c-b-α)e=0 for some α ∈ C and f(x₁,…,x_n) is central-valued in eRe.

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G₁→ G₀→ G_-1→ ⋯ such that the complex Hom_R(H,G) is exact for each Gorenstein injective R-module H, is the module Im(G₀→ G_-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring.

Let R be a commutative Noetherian ring, I, J two ideals of R, and M an R-module. For a non-negative integer t, we show: (a) If for all j < t and are finite (resp., Artinian), then is finite (resp., Artinian). (b) If for all j < t and are finite (resp., Artinian), then is finite (resp., Artinian). In addition, if (R,𝔪) is a local ring, J a non-nilpotent ideal, and M a finite R-module, then we show that is not Artinian for some i ∈ ℕ₀.

In this paper we investigate necessary and sufficient conditions on an ideal under which it is an exchange ideal with all idempotents central. Related examples are constructed by means of ideal-extensions. A kind of uniqueness in such ideals is also studied. These provide a new class of such ideals.

Let (R,𝔪) be a Cohen-Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. Let a₁,…,a_d be a joint reduction of (I^[d-1]|K), and set L=(a₁,…,a_d), J=(a₁,…,a_d-1). When depth G(I) ≥ d-1 and depth F_K(I) ≥ d-2, we show that the lengths , and the joint reduction number r_L(I|K) are independent of L. In the general case, we give an upper bound of the Hilbert series of F_K(I). When depth G(I) ≥ d-1, we also provide a characterization, in terms of the Hilbert series of F_K(I), of the condition depth F_K(I) ≥ d-1.

Let k be a positive integer. Locally graded groups G for which one of the sets {γ_k(H)| H ≤ G} and {γ_k(H)| H ≤G, H infinite} is finite are classified.

Let (R,𝔪) be a Noetherian local ring and M a finitely generated R-module. For an integer i ≥ 0, the Artinian i-th local cohomology module is said to satisfy the shifted localization principle if

In this paper, through the calculation of K₂(𝔽_pC_p²[t]/(t^p2),(t)) using Dennis-Stein symbols, we determine the explicit structure of K₂(𝔽_p[C_p² × C_p²]) and give its basis in Steinberg symbols.

Let H be a finite dimensional Hopf algebra and A be an algebra over a fixed field k. Firstly, it is proved that the left FP-projective dimension is invariant under cleft extensions when H is semisimple and A is left coherent. Secondly, using (co)induction functors, we study the relations between FP-projective dimensions in A # H-Mod and the counterparts in A^H-Mod. Finally, we characterize the FP-projective preenvelopes (resp., precovers) under H*-extensions and cleft extensions, respectively.

As an extension of the basic model of MRA-codes, multi-receiver multi-fold authentication codes can use a single key distribution phase for multiple message transmission. A multi-receiver multi-fold authentication code is constructed from the singular symplectic geometry over finite fields in this paper. The parameters and probabilities of success in impersonation and substitution attacks by malicious groups of receivers of this code are computed.

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, f(x₁,…,x_n) a multilinear polynomial over C which is not an identity for R, F and G two non-zero generalized derivations of R. If F(u)G(u)=0 for all u ∈ f(R)= {f(r₁,…,r_n): r_i∈ R}, then one of the following holds: (i) There exist a, c ∈ U such that ac=0 and F(x)=xa, G(x)=cx for all x ∈ R; (ii) f(x₁,…,x_n)² is central valued on R and there exist a, c ∈ U such that ac=0 and F(x)=ax, G(x)=xc for all x ∈ R; (iii) f(x₁,…,x_n) is central valued on R and there exist a,b,c,q ∈ U such that (a+b)(c+q)=0 and F(x)=ax+xb, G(x)=cx+xq for all x ∈ R.