Journal of Algebra and Its Applications

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module _RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If s_n is a standard skew-symmetric polynomial of degree n, then any s₄-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D²) are s₄-Lie. An algebra (U, id ∧ D³ - 2D ∧ D²) is s'₅-Lie, where s'₅ is a non-standard skew-symmetric polynomial of degree 5.

A ring is called n-perfect (n ≥ 0), if every flat module has projective dimension less or equal than n.

In this paper, we show that the n-perfectness relates, via homological approach, some homological dimensions of rings. We study n-perfectness in some known ring construction. Finally, several examples of n-perfect rings satisfying special conditions are given.

The notions of annihilator ideal and split structure are studied in detail for both, dialgebras and quasi-Jordan algebras. It yields methods for additional units in the two structures. As a consequence the notion of regular element receives special attention.

This paper generalize the idea of the authors in J. Pure Appl. Algebra210 (2007) 437–445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.

A subgroup H of G is said to be self-conjugate-permutable if HH^x = H^xH implies H^x = H. A finite group G is called PSC-group if every cyclic subgroup of group G of prime order or order 4 is self-conjugate-permutable. In the paper, first we give the structure of finite group G, all of whose maximal subgroups are PSC-groups. Then we also classified that finite group G all of whose second maximal subgroups are PSC-groups.

We associate a graph to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x,y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever and the equality for a non-solvable group G holds if and only if G/Cyc(G) ≅ A₅ or S₅.

We determine the regular representations, Gabriel quivers and representation type of all liftings of two-dimensional quantum linear spaces.