Journal of Algebra and Its Applications

This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈ Θ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras H₁ and H₂ are called geometrically equivalent if and only if the H₁-closed sets coincide with the H₂-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H₁ and H₂ are called automorphically equivalent if and only if the H₁-closed sets coincide with the H₂-closed sets after some "changing of coordinates". We can detect the difference between geometric and automorphic equivalence of algebras of the variety Θ by researching of the automorphisms of the category Θ⁰ of the finitely generated free algebras of the variety Θ. By [5] the automorphic equivalence of algebras provided by inner automorphism coincide with the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group 𝔄/𝔜 is big. Here 𝔄 is the group of all automorphisms of the category Θ⁰, 𝔜 is a normal subgroup of all inner automorphisms of the category Θ⁰. In [6] the variety of all Lie algebras and the variety of all associative algebras over the infinite field k were studied. If the field k has not nontrivial automorphisms then group 𝔄/𝔜 in the first case is trivial and in the second case has order 2. We consider in this paper the variety of all linear algebras over the infinite field k. We prove that group 𝔄/𝔜 is isomorphic to the group (U(kS₂)/U(k{e}))λAutk, where S₂ is the symmetric group of the set which has 2 elements, U(kS₂) is the group of all invertible elements of the group algebra kS₂, e ∈ S₂, U(k{e}) is a group of all invertible elements of the subalgebra k{e}, Autk is the group of all automorphisms of the field k. So even the field k has not nontrivial automorphisms the group 𝔄/𝔜 is infinite. This kind of result is obtained for the first time. The example of two linear algebras which are automorphically equivalent but not geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.

An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G′. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) ≃ Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner.

In this paper, we investigate the structure of finite groups by means of normally embedded subgroups.

In this paper, we study some properties of generalized Krull domains. Among other things, we give several characterizations of generalized Krull domains.

In this paper, we investigate sufficient conditions which ensure that the minimal number of generators of a toric ideal I_A is equal to its binomial arithmetical rank. We study particularly the case where I_A is the toric ideal of a graph.

We obtain an improvement on the Casolo and Dolfi's result about ρ – σ conjecture for irreducible character degrees.

In this paper, we classify finite 2-groups with a unique minimal non-abelian subgroup of index 2. We also classify finite 2-groups with a 2-generator abelian subgroup of index 2. This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p. Together with our previous work, we solved completely a problem proposed by Y. Berkovich.

We determine the irreducible constituents of the Steinberg character of an orthogonal group over a finite field restricted to the orthogonal group of one less dimension.

We consider the branching problem for generalized Verma modules M_λ(𝔤, 𝔭) applied to couples of reductive Lie algebras . Our analysis of the problem is based on projecting character formulas to quantify the branching, and on the action of the center of to construct explicitly singular vectors realizing the -top level of the branching. We compute explicitly the top part of the branching for the pair for both strongly and weakly compatible with i(LieG₂) parabolic subalgebras and a large class of inducing representations.

In this paper, we define and study the cyclic graph Γ_S of a finite semigroup S. We obtain some graph theoretical properties of Γ_S including its dominating number, independence number and genus of the graph. Then, we study the cyclic graphs Γ_ℤn and Γ_GLn(F), where ℤ_n is the semigroup under multiplicative operation and GL_n(F) is the set of all n × n invertible matrices over a finite field F with multiplicative operation.

We define a "mirror version" of Brzeziński's crossed product and we prove that, under certain circumstances, a Brzeziński crossed product D ⊗_R,σ V and a mirror version may be iterated, obtaining an algebra structure on W ⊗ D ⊗ V. Particular cases of this construction are the iterated twisted tensor product of algebras and the quasi-Hopf two-sided smash product.

It is shown that RgMax(R) is infinite for certain commutative rings, where RgMax(R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then |RgMax(R)| ≥ |Irr(D) ∩ U(R)|, where Irr(D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or |RgMax(R)| ≥ ℵ₀. It is proved that if R is a zero-dimensional (or semilocal) ring with |RgMax(R)| < ℵ₀, then R has nonzero characteristic, say n, and R is integral over ℤ_n. In particular, it is shown that if R is an uncountable artinian ring, then |RgMax(R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2^ℵ₀, then |RgMax(R)| ≥ 2^ℵ₀. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R₂ ⊂ R₁ ⊂ R₀ = R where each R_i is a maximal subring of R_i-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules.

Isomorphy classes of k-involutions have been studied for their correspondence with symmetric k-varieties, also called generalized symmetric spaces. A symmetric k-variety of a k-group G is defined as G_k/H_k where θ : G → G is an automorphism of order 2 that is defined over k and G_k and H_k are the k-rational points of G and H = G^θ, the fixed point group of θ, respectively. This is a continuation of papers written by A. G. Helminck and collaborators [Involutions of SL(2, k), (n > 2), Acta Appl. Math.90(1–2) (2006) 91–119, Classification of involutions of SO(n; k; b), to appear, On the classification of k-involutions, Adv. Math.153(1) (1988) 1–117, Classification of involutions of SL(2, k), Comm. Algebra30(1) (2002) 193–203] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type A, B and D. Here we begin a series of papers doing the same for algebraic groups of exceptional type.

The power graph of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph of is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph and its subgraph . We next provide necessary and sufficient conditions for a power graph to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph contains some cut-edges.

In this paper, we introduce the notion of a Wakamatsu tilting pair as a common generalization of a Wakamatsu tilting module and a tilting pair. We investigate some properties of Wakamatsu tilting pairs, and construct a Wakamatsu tilting right End_AC-module from a given Wakamatsu tilting pair (C, T). And some characterizations of Wakamatsu tilting pairs are given in terms of subcategories.

Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In this paper, three more new criteria are given.

In this paper, we consider aspects of the big lattice of preradicals, related to pseudocomplements and supplements. We consider essential preradicals and superfluous preradicals, and we characterize the situation in which all nonzero preradicals are essential as well as the one in which all proper preradicals are superfluous. Also, we give two different proofs of the known fact that the big lattice of preradicals is strongly pseudocomplemented.

Let K be a field of characteristic p > 0 and let G be an arbitrary group. In this paper, we classify group algebras KG which are strongly Lie nilpotent of index at most 8. We also show that for k ≤ 6, KG is strongly Lie nilpotent of index k if and only if it is Lie nilpotent of index k.

In this paper, we show that every locally graded group with a finite number k of derived subgroups is nilpotent-by-abelian-by-(finite of order ≤ k!)-by-abelian. Also we give some sufficient conditions on the number of derived subgroups of a locally graded group to be solvable.

We find natural numbers m such that the dimensions of global sections of multiple adjoint bundles h⁰(m(K_X + L)) are strictly greater than zero for any quasi-polarized n-folds (X, L) for which X is a complex normal Gorenstein projective variety of dimension n with only rational singularities and K_X + L is nef.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc(G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Let R be a ring, S a strictly ordered monoid, and ω : S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and generalized power series rings. We characterize those subsets T of S for which the cut-off operator with respect to T is a Rota–Baxter operator on the ring R[[S, ω]]. The obtained results provide a large class of noncommutative Rota–Baxter algebras.

A classical theorem of J. L. Burchnall and T. W. Chaundy shows that two commuting differential operators P and Q give rise, via a differential resultant, to a complex algebraic curve with equation F(x, y) = 0, such that formally inserting P and Q for x and y in F(x, y), gives identically zero. In addition, the points on this curve have coordinates which are exactly the eigenvalues associated with the operators P and Q (see the Introduction for a more precise statement). In this paper, we prove a generalization of this result using resultants in Ore extensions.