Let $R$ be a ring, and let $M$ be a right $R$-module. Recently, Baannon and Khalid introduced and studied the concept of $e^*$-essential submodules, extending the notion of essential submodules. In this context, we consider $e^*$-semisimple submodules, and we define the $e^*$-socle of $M$ as the largest $e^*$-semisimple submodule of $M$. It turns out that several properties of the socle can be extended to the $e^*$-socle. It is shown that the $e^*$-socle of $M$ is exactly the intersection of all its $e^*$-essential submodules.
| Mualliflar | Bouziri , Ayoub |
|---|---|
| Jurnal | O‘zbekiston matematika jurnali |
| Nashr sanasi | 2025-11-03 |
| Jild | 69 |
| Son | 4 |
| Betlar | 46-52 |
| Til | Ingliz |
| DOI | 10.29229/uzmj.2025-4-6 |
DOI: 10.29229/uzmj.2025-4-6 · Maqolaning asl sahifasi
$e^*$-essential submodules, $e^*$-semisimple modules, socle, $e^*$-socle
Let $Q$ be the field of rational numbers and $Q^{2}$ be the $2$-dimensional linear space over $Q$. A classification of all non-degenerate symmetric bilinear-metric forms over $Q^{2}$ have obtained. Let $\varphi $ be a…
In this paper, we classify the symmetric Leibniz algebras whose underlying Lie algebra is almost filiform. We also describe the symmetric Leibniz algebras associated with Heisenberg and triangular Lie algebras…
Let $B$ be a complete Boolean algebra, let $Q(B)$ be the Stone compact of $B$, let $C_\infty (Q(B))$ be the commutative unital algebra of all continuous functions $x:Q(B) \rightarrow [-\infty,+\infty],$…
The time-optimal problem is considered for the controllable heat conduction process in a rod. Using the Fourier method, the problem is usually reduced to an infinite system of one-dimensional control equations whose…
In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a…
In this paper, we first classify all associative algebra structures on a two-dimensional vector space over an arbitrary base field equipped with a non-degenerate bilinear form. We then determine which of these are…
Hypergeometric functions are divided into complete and confluent functions. For the first time, Srivastava and Karlsson described a method for constructing a set of all possible triple Gaussian hypergeometric series and…
In this work, the problem of stabilizing the equilibrium state for a hyperbolic system with negative nonlocal characteristic velocities and measurement error is investigated. A mixed problem is considered for such…
Hamilton systems play a fundamental role in mechanics and mathematics, which usually generated by Hamiltonian vector fields.The geometry and topology of vector field orbits is not only an important branch of modern…
In the paper, we consider a class dynamic control system with a discrete parameter and under conditions of uncertainty in the initial data. The optimal control problem of the minimax type is formulated for a non-smooth…