Galois Journal of Algebra

Abstract This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and finite sequences of such algebras over fields whose characteristic divides the degree of the algebras under discussion. Upper and lower bounds as well as a few precise evaluations of this dimension are included.

Abstract This study aims to investigate the low dimensional complex Leibniz algebras which admit a periodic
derivation. The principal goal of this note is to characterize such algebras and to develope some properties
on periodic derivations. We demonstrate that finite dimensional complex Leibniz algebras admitting a
periodic derivation are abelian or at most 2-class nilpotent. Moreover, we prove that the order of a
periodic derivation in such algebras is divided by 6.

Abstract Due to their elegant and simple nature, unitary Cayley graphs have been an active research topic in the literature. These graphs are naturally connected to several branches of mathematics, including number theory, finite algebra, representation theory, and graph theory. In this article, we study the perfectness property of these graphs. More precisely, we provide a complete classification of perfect unitary Cayley graphs associated with finite rings.

Abstract In this survey we give a brief account of the history of pseudo-linear maps since their introduction by N. Jacobson in 1937.
The ($\si,\de$)-pseudo-linear transformations are introduced via the modules over a skew polynomial ring.
Many classical properties of linear maps are shown to have analogues in the setting of ($\si, \delta$)-PLT's.
The relations between these maps and the evaluation of skew polynomials is particularly
emphasized. It is shown, in particular, how they are useful while evaluation inside a conjugacy class is considered. The evaluation of Ore polynomials with more then one variables is shown to be related to sequence of pseudo-linear maps. We also briefly present a recent application that allows to easily answer an open question. Many examples are presented all along the text.

Abstract This survey presents a comprehensive and systematic overview of the theory of chain conditions on modules and rings with particular emphasis on injective and quasi-injective modules, as well as associated structural properties of rings. We cover classical and modern perspectives on ascending and descending chain conditions (acc and dcc) on submodules, essential submodules, and quotient modules. Key classes such as Noetherian, Artinian, seminoetherian, and isonoetherian modules and rings are explored in detail. We review the theory of quasi-injective modules under chain conditions, including characterization
theorems, endomorphism ring decompositions, and their connections to quasi-Frobenius rings. The notions of essential Noetherian and essential Artinian modules and their significance in generalizing classical results. Finally, we touch upon divisibility properties in submodule chains, power series ring extensions, projective modules over semilocal rings, and related topics, highlighting open problems and future research directions.

Abstract In this survey, the author collects and presents several of his recent results and joint works on monomorphism categories, covering various topics in representation theory. We try to show that how one can approach the monomorphism categories via functor categories. In the representation-finite case, we are essentially dealing with module categories, so well-understood knowledge from module categories can be transferred to monomorphism categories. In particular, this approach is effective for developing covering theory for monomorphism categories. The results in Section~3, which provide various equivalences between the monomorphism category and the category of finitely presented covariant functors, are taken from an unpublished paper by the author.

Abstract We investigate two parametric families of irreducible sextic polynomials over $\mathbb{Q}$, denoted $g(x;t)$ and $h(x;t)$, and the number fields they generate. For integers $t$ in suitable ranges, we show that the fields
\[
K_t = \mathbb{Q}(\alpha) \quad \text{and} \quad L_t = \mathbb{Q}(\theta),
\]
where $\alpha$ and $\theta$ are roots of $g(x;t)$ and $h(x;t)$, respectively, exhibit rich arithmetic and algebraic structure.

In particular, both families define exceptional number fields, and we prove that for infinitely many $t$, the fields are monogenic. We also show that $K_t$ contains real quadratic subfields of the form
\[
\mathbb{Q}(\sqrt{t^2 - 4}),
\]
and that every real quadratic field embeds in some $K_t$. Meanwhile, each $L_t$ contains a cubic subfield of the form
\[
\mathbb{Q}(\theta^2 - \theta).
\]

These results suggest that exceptional and monogenic number fields arise naturally and frequently in well-structured parametric families.