Original Article
Essential ideal transforms

Ismael Akray; Runak Mustafa

Volume 1, Issue 1 , December 2025, Pages 1-11

Abstract
  It is our intention in this research generalize some concept in local cohomology such as free modules, contravariant functor $ext$, covariant functor $Ext$ and ideal transforms with $e$-exact sequences. The $e$-exact sequence was introduced by Akray and Zebari \cite{AZ} in 2020. we prove that essential ...  Read More

Review Article
Some Perspectives on Distribution of strongly persistent clutters

Jonathan Toledo Toledo; Enrique Reyes Espinoza

Volume 1, Issue 1 , December 2025, Pages 12-42

Abstract
  This survey aims to present and promote the study of the strong persistence property in clutters from a quantitative and structural perspective. We denote by Cn the set of all clutters on [n], by Sn the subset of clutters satisfying the SPP, and by Nn = Cn ∖ Sn those that fail it. Our analysis focuses ...  Read More

Original Article
A family of rank 4 non-algebraic matroids with pseudomodular dual

Winfried Hochstättler

Volume 1, Issue 1 , December 2025, Pages 43-50

Abstract
  The Tic-Tac-Toe matroid is a paving matroid of rank 5 on 9 ele-ments which is pseudomodular and whose dual is non-algebraic. It has beenproposed as a possible example of an algebraic matroid whose dual is not algebraic. We present an infinite family of matroids sharing these properties andgeneralizing ...  Read More

Review Article
Recent Advances in the Theory of Polyomino Ideals

Ayesha Asloob Qureshi; Francesco Navarra

Volume 1, Issue 1 , December 2025, Pages 51-87

Abstract
  Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a unified account of recent advances on two central themes: the ...  Read More

Review Article
Advances in Arithmetic and Geometry over division rings

Elad Paran

Volume 1, Issue 1 , December 2025, Pages 88-117

Abstract
  In recent years there has been a flurry of activity surrounding non-commutative Nullstellens\"atze over the real quaternion algebra and over general division rings. From these results emerge rich arithmetic and geometry, which in many ways follow the classical themes and ideas of complex algebraic geometry, ...  Read More

Original Article
Classification, derivations and centroids of low-dimensional associative trialgebras

Bouzid Mosbahi; Imed Basdouri; Ahmed Zahari

Volume 1, Issue 1 , December 2025, Pages 118-136

Abstract
  In this paper, we study the structure and algebraic varieties of associative trialgebras. In partic-ular, we classify all associative trialgebras of dimension at most four over a field of characteristic zero. Based on this classification, we provide a detailed analysis of their derivations and centroids. ...  Read More