Journal of Algebra and Its Applications

We establish several characterizations of maximal non-Prüfer and maximal non-integrally closed subrings of a field. Special attention is given when finiteness conditions are satisfied. Several numerical characterizations are then obtained. The paper is concluded with examples that exhibit the obtained results.

A module M is called a dimension module if the Goldie (uniform) dimension satisfies the formula u(A + B) + u(A ∩ B) = u(A) + u(B) for arbitrary submodules A, B of M. Dimension modules and related notions were studied by several authors. In this paper, we study them in a more general context of modular lattices with 0 to which the notion of dimension modules can be extended in an obvious way. Some constructions available in the lattice theory framework make it possible to identify several new aspects concerning the nature of dimension lattices and modules as well as to describe a number of related properties. In particular we find a lattice which can be used to test whether a given lattice or a module satisfies the studied properties. Most of the results are obtained for lattices and then they are applied to modules. However the examples are given, when possible, in the more restrictive case of modules.

We show that associative systems with a sufficiently good module structure imbed in a primitive system with simple primitive heart, spanned by the original system and the heart, so extending the results of J. Pure Appl. Algebra 181 (2003) 131–139 to systems over more general rings of scalars. We also study associative systems with involution.

Let R be a commutative unital ring of arbitrary characteristic and let G be a multiplicative Abelian group. For the group ring RG we completely calculate the number (finite or infinite) of its idempotents only in terms of R, G and their sections. This strengthens our previous results in Sarajevo J. Math. (2011) and Filomat (2012).

In the present paper we extend the results of [2, 4] for the tensor square of Lie algebras. More precisely, for any Lie algebra L with L/L² of finite dimension, we prove L ⊗ L ≅ L □ L ⊕ L ∧ L and Z^∧(L) ∩ L² = Z^⊗(L). Moreover, we show that L ∧ L is isomorphic to derived subalgebra of a cover of L, and finally we give a free presentation for it.

Let π be a group and H be a Hopf π-coalgebra. We investigate the criterion for the existence of a total integral of π-H-comodule algebra A in the setting of Hopf π-coalgebras, and prove that there exists a total integral θ = {θ_α : H_α → A} if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of H, which generalizes the result invented by Doi in the ordinary Hopf algebra setting.

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

We obtain a bound on the number of solutions of x^q = x in a finite noncommutative algebra over a field with q elements. Furthermore, we completely characterize those rings for which this maximum number is attained.

It is shown that if a bilinear map f : A × B → C of modules over a commutative ring k is nondegenerate (i.e. if no nonzero element of A annihilates all of B, and vice versa), and A and B are Artinian, then A and B are of finite length.

Some consequences are noted. Counterexamples are given to some attempts to generalize the above result to balanced bilinear maps of bimodules over noncommutative rings, while the question is raised whether other such generalizations are true.

We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic p ≥ 5.

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting L_e(G) = {x ∈ G|x^e = 1}. Frobenius proved that |L_e(G)| = ke for some positive integer k ≥ 1. Let k(G) be the upper bound of the set {k||L_e(G)| = ke, ∀ e ||G|}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.

In this paper, we first discuss the nilpotency of restricted Lie triple systems and the condition of existence of p-mappings on Lie triple systems. Second, we devote our attention to prove the uniqueness of the decomposition as a direct sum of p-ideals of a restricted Lie triple system. Finally, we study how a quasi-toral restricted Lie triple system T with zero center and of minimal dimension should be.

A half-isomorphism φ : G → K between multiplicative systems G and K is a bijection from G onto K such that φ(ab) ∈ {φ(a)φ(b), φ(b)φ(a)} for any a, b ∈ G. It was shown by W. R. Scott [Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if G is a group then φ is either an isomorphism or an anti-isomorphism. This is used to prove that a finite group is determined by its group determinant. Here we show that every half-isomorphism between Moufang loops of odd order is either an isomorphism or an anti-isomorphism. Such a result does not carry over to Moufang loops of even order.

We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [A. Aubert, Structures affines et pseudo-métriques invariantes à gauche sur des groupes de Lie, Thèse, Université Montpellier II (1996)]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608]. We study the structures of even (respectively, odd)-quadratic odd (respectively, even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superalgebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608].

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

In this paper, we classify the left-symmetric superalgebras in dimensions two and three over the field of complex numbers. As a direct consequence, we obtain some explicit solutions of the graded classical Yang–Baxter equation in certain Lie superalgebras and their corresponding Lie bisuperalgebras.

In this paper we provide a characterization of Lie solvable group algebras of derived length three over a field of characteristic three when G is a non-2-Engel group with abelian commutator subgroup.

Let R be a right pure semisimple ring, and be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory in Mod-R determined by the family , and show that has several nice properties similar to those of the category Mod-R. For example, its functor category is a module category, and preinjective objects of are sources of left almost split morphisms in and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime p, denote by the class of all 2-generated commutative automorphic loops Q possessing a central subloop Z ≅ ℤ_p such that Q/Z ≅ ℤ_p × ℤ_p. Upon describing the free 2-generated nilpotent class two commutative automorphic loop and the free 2-generated nilpotent class two commutative automorphic p-loop F_p in the variety of loops whose elements have order dividing p² and whose associators have order dividing p, we show that every loop of is a quotient of F_p by a central subloop of order p³. The automorphism group of F_p induces an action of GL₂(p) on the three-dimensional subspaces of Z(F_p) ≅ (ℤ_p)⁴. The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from . We describe the orbits, and hence we classify the loops of up to isomorphism.

It is known that every commutative automorphic p-loop is nilpotent when p is odd, and that there is a unique commutative automorphic loop of order 8 with trivial center. Knowing up to isomorphism, we easily obtain a classification of commutative automorphic loops of order p³. There are precisely seven commutative automorphic loops of order p³ for every prime p, including the three abelian groups of order p³.

We give necessary and sufficient conditions for the functor F from the category of partial entwined modules to the category of right A-modules to be separable. This leads to a generalized notion of integrals of partial entwining structures. As an application, we prove a Maschke-type theorem for partial entwined modules.

We characterize the abelian groups G for which Ext(G, -) commutes with direct sums.