Journal of Algebra and Its Applications

Let R be a ring, R is called a right ACS ring if the right annihilator of each element is essential in a direct summand of R. We prove that right CF ring is right artinian under right ACS condition. In particular, FGF conjecture is true under right ACS condition. We also give some conditions under which right noetherian left P-injective and left ACS ring is QF.

Let k be a perfect field, K its algebraic closure and let Λ be a finite-dimensional k-algebra. A result of Kasjan asserts that if k is infinite and Λ is generically tame then Λ ⊗_k K is generically tame. We show that the infiniteness requirement can be removed.

In this paper, we introduce the Cayley graph of a partially ordered set (poset). Let (P, ≤) be a poset, and let S be a subset of P. We define the undirected Cayley graph of P, denoted by Cay(P, S), as a graph with vertex-set P and edge-set E consisting of those sets {x, y} such that y ∈ {x, s}^ℓ or x ∈ {y, s}^ℓ for some s ∈ S, where for a subset T of P, T^ℓ is the set of all x ∈ P such that x ≤ t, for all t ∈ T. We study some basic properties of Cay(P, S) such as connectivity, diameter and girth.

We propose a decoding algorithm for the (u | u + v)-construction that decodes up to half of the minimum distance of the linear code. We extend this algorithm for a class of matrix-product codes in two different ways. In some cases, one can decode beyond the error-correction capability of the code.

We study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.

We prove that if any cyclic 2′-subgroup of a finite group G centralizes all real 2-elements of order ≤ 4, then G is 2-nilpotent (Theorem 1). Some related results are proved. Minimal nonnilpotent groups and minimal nonabelian 2-groups play important role in our proof of those results.

We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring.

A subgroup H of a finite group G is called ss-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is s-quasinormal in K. In this paper, we investigate the influence of ss-supplemented minimal subgroups on the structure of finite groups and obtain some interesting results.

In this paper, we consider a tower of function fields over a finite field 𝔽_q and a finite extension E/F₀ such that the sequence is a tower over the field 𝔽_q. Then we study invariants of , that is, the asymptotic number of the places of degree r in , for any r ≥ 1, if those of are known. We first give a method for constructing towers of function fields over any finite field 𝔽_q with finitely many prescribed invariants being positive. For q a square, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions. Moreover, we show the existence of towers over a finite field 𝔽_q attaining the Drinfeld–Vladut bound of order r, for any r ≥ 1 with q^r a square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A₂ by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras with Dynkin diagrams of type A, D, or E, except for A₁ and A₁ × A₁ with the order N_J > 2 for at least one component J, are wild.

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id_R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id_R, we show that it intersects every nonzero ideal of R[x;id_R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id_R, δ], then R[x;id_R, δ] is simple. We also show that under some conditions on R the converse holds.

Let G be a finite group and H be a subgroup of G. We introduce the non-normal graph of H in G, denoted by _H,G, and give some of the graph theoretical properties of _H,G.

A new criteria for the solvability of finite groups is obtained. It is proven that a finite group G is always solvable if

(i) H is subnormal in G or |N_G(H) : H| ≤ 2 for every non-cyclic 2-subgroup H of G that is not a subgroup of Z(G), or

(ii) H is subnormal in G or G is A₄-free and |N_G(H) : H| ≤ 6 for every non-cyclic 2-subgroup H of G that is not a subgroup of Z(G).

Moreover, some new criteria for a finite group being p-nilpotent are also obtained.

In this paper, we compute minimal faithful representations of filiform Lie algebras by means of strictly upper-triangular matrices. To obtain such representations, we use nilpotent Lie algebras _n, of n × n strictly upper-triangular matrices, because any given (filiform) nilpotent Lie algebra admits a Lie-algebra isomorphism with a subalgebra of _n for some n ∈ ℕ\{1}. In this sense, we search for the lowest natural integer n such that the Lie algebra _n contains the filiform Lie algebra as a subalgebra. Additionally, we give a representative of each representation.

Let (R,) be a commutative Noetherian local ring and let M and N be nonzero finitely generated R-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper, we prove a generalization of Ischebeck formula, that is .

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by T_Γ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of T_Γ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in T_Γ(R) is denoted by I_TΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph I_TΓ (ℤ_n) of gamma sets in T_Γ(ℤ_n). In this paper, we study about I_TΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of I_TΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam(I_TΓ(R)) ≤ 2 and gr(I_TΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of I_TΓ(R) are given. Further, we discuss about the vertex-transitive property of I_TΓ(R). At last, we obtain all commutative Artin rings R for which I_TΓ(R) is either planar or toroidal or genus two.

The intersection graph I_TΓ(R) of gamma sets in the total graph T_Γ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about I_TΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on I_TΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of I_TΓ(R). Further, we focus on certain graph theoretic parameters of I_TΓ(R) like the independence number, the clique number and the connectivity of I_TΓ(R). Also, we obtain both vertex and edge chromatic numbers of I_TΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(I_TΓ(R)) = ω(I_TΓ(R)). Having proved that I_TΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which I_TΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which I_TΓ(R) is of class one (i.e. χ′(I_TΓ(R)) = Δ(I_TΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of I_TΓ(R) are equal to the degree of any vertex in I_TΓ(R).

Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.

Two special types of module subcategories are defined over stratified algebras of Cline, Parshall and Scott. We show that for every stratified algebra there exists a (not necessarily unique) cotorsion pair of subcategories which describe to a large extent the stratification structure of the algebra. These subcategories generalize the notion of modules with standard and costandard filtration for standardly stratified and quasi-hereditary algebras.

In this paper we consider partial actions of groups on semiprime rings. Our purpose is to study when such a partial action has a globalization.

We study top generalized local cohomology modules and get important properties about the vanishing, non-vanishing and the artinianness of generalized local cohomology modules.

For a finite group G let σ(G) (the "sum" of G) be the least number of proper subgroups of G whose set-theoretical union is equal to G, and σ(G) = ∞ if G is cyclic. We say that a group G is σ-elementary if for every non-trivial normal subgroup N of G we have σ(G) < σ(G/N). In this article we produce the list of all the σ-elementary groups of sum up to 25. We also show that σ(Aut(PSL(2, 8))) = 29.