Journal of Algebra and Its Applications

Let A be the elementary group of order 2ⁿ and L an A-graded Lie algebra with L₀ = 0. Assume that L is soluble with derived length k. It is proved that L has a series of ideals of length n all of whose quotients are nilpotent of {k, n}-bounded class.

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x₁, …, x_n]. Let J(R) be the ideal in 𝕜[X] defined by . It is shown that if it is a principal ideal then , where q = pⁿ(p - 1)/2.

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GL_n(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.

Given a ring R, we study the bimodules M for which the trivial extension R ∝ M is morphic. We obtain a complete characterization in the case where R is left perfect, and we prove that R ∝ Q/R is morphic when R is a commutative reduced ring with classical ring of quotients Q. We also extend some known results concerning the connection between morphic rings and unit regular rings.

The structure of the unit group of the group algebra of the group C₂ × D₈ over any field of characteristic 2 is established in terms of split extensions of cyclic groups.

Let A be a finite-dimensional hereditary algebra over an algebraically closed field k, A^(m) be the m-replicated algebra of A and be the m-cluster category of A. In this paper, we introduce the notion of mutation team in mod A^(m), and prove that each faithful almost complete tilting module over A^(m) has a mutation team by showing that the sequence of the complements satisfies the properties of the mutation team. We also prove that for each partial mutation team in the m-left part of mod A^(m), there exists a faithful almost complete tilting module having the partial mutation team as the set of indecomposable complements. As an application, we prove that m-cluster mutation in can be realized as tilting mutation in mod A^(m), and we also give the relationship between connecting sequences in mod A^(m) and higher AR-angles in the m-cluster category .

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let C_G(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph K_p-1, a vertex c ∈ C_G(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (K_p)).

The number of ways an element of a finite group can be expressed as a square, a commutator, or more generally in the form w(x₁, …, x_r), where w is a word in the free group, defines a natural class function. We investigate some properties of these class functions, in particular their tendency to be characters or virtual characters of the underlying group. Generalizing classical results of Frobenius and others, we prove that generalized commutators yield characters in this manner, and use this to exhibit a criterion for nilpotency based on a certain equation associated with the irreducible characters.

An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f^-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte–Sfax–Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669–674."

This paper studies the integral and complete integral closures of an ideal in an integral domain. By definition, the integral closure of an ideal I of a domain R is the ideal given by I′ ≔ {x ∈ R | x satisfies an equation of the form x^r + a₁x^r-1 + ⋯ + a_r = 0, where a_i ∈ Iⁱfor each i ∈ {1, …, r}}, and the complete integral closure of I is the ideal Ī ≔ {x ∈ R | there exists 0 ≠ = c ∈ R such that cxⁿ ∈ Iⁿfor all n ≥ 1}. An ideal I is said to be integrally closed or complete (respectively, completely integrally closed) if I = I′ (respectively, I = Ī). We investigate the integral and complete integral closures of ideals in many different classes of integral domains and we give a new characterization of almost Dedekind domains via the complete integral closure of ideals.

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (u_{k, m}(x), u_{k, m}(x*)) or (u_{k, m}(x), u_{k, m}(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam(𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr(𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with 𝔸(R) be its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph 𝔸𝔾(R) that its vertices are 𝔸(R)* = 𝔸(R)\{(0)} in which for every distinct vertices I and J, I — J is an edge if and only if IJ = (0). First, we study the diameter of 𝔸𝔾(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(𝔸𝔾(R)) ≤ 2 or R is reduced and χ(𝔸𝔾(R)) ≤ ∞. Also it is shown that for each reduced ring R, χ(𝔸𝔾(R)) = cl(𝔸𝔾(R)). Moreover, if χ(𝔸𝔾(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(𝔸𝔾(R)) = cl(𝔸𝔾(R)) = n. Finally, we show that for a Noetherian ring R, cl(𝔸𝔾(R)) is finite if and only if for every ideal I of R with I² = (0), I has finite number of R-submodules.

Let A be an integral domain. We study new conditions on families of integral ideals of A in order to get that A is of t-finite character (i.e. each nonzero element of A is contained in finitely many t-maximal ideals). We also investigate problems connected with the local invertibility of ideals.

For iterated Ore extensions satisfying a polynomial identity (PI) we present an elementary way of erasing derivations. As a consequence we recover some results obtained by Haynal. [PI degree parity in q-skew polynominal rings, J. Algebra319 (2008) 4199–4221]. We also prove that under mild assumptions on R_n = R[x₁; σ₁, δ₁]⋯ [x_n;σ_n;δ_n], the Ore extension R[x₁;σ₁]⋯[x_n;σ_n] exists and is PI if R_n is PI.

Let R be a commutative regular ring of dimension d, I be an ideal of R and M be a non-zero finitely generated R-module. In this paper we show that Huneke's two different conjectures are equivalent. Also we provide some partially affirmative answers to them. In fact it is shown that the Bass numbers of are finite for all i ≥ 0, whenever d ≤ 3. Also if (R, m) be regular local ring, we show that the Bass numbers are finite, for all i ≥ 0 and all j ≥ 0, and , for all i ≥ 0 and all j ≥ 0, whenever height(I) = 1.

When R is a commutative ring with identity, and if k ∈ ℕ, with kR = R, then it was shown in [C. Weibel, Mayer–Vietoris Sequence and Module Structure on NK₀, Lecture Notes in Mathematics, Vol. 854 (Springer, 1981), pp. 466–498] that SK₁(R[X]) has no k-torsion. We prove this result for any associative ring R with identity in which kR = R.