This study delves into the properties of the associated act V over the monoid S of sinshT. It examines the relationship between faithful, finitely generated, and separated acts, as well as their connections to one-to-one and onto operators. Additionally, the correlation between acts over a monoid and modules over a ring is explored. Specifically, it is established that  functions as an act over S if and only if  functions as module, where T represents a nilpotent operator. Furthermore, it is proved that when T is onto operator and  is finitely generated, is guaranteed to be finite-dimensional. Prove that for any bounded operator the following,  is acting over S if and only if  is a module where T is a nilpotent operator, is a

 الأثر V بالنسبة إلى   sinshT و خواصه قد تم دراسته في هذا البحث حيث تم دراسة علاقة الأثر المخلص والاثر المنتهى التولد والاثر المنفصل وربطها بالمؤثرات المتباينة حيث تم بهنة العلاقات التالية ان الاثر اذا وفقط اذا مقاس في حالة كون المؤثر هو عديم القوة وكذلك في حالة كون المؤثر شامل فان الاثر هو منتهي التولد اي ان الغضاء هو منتهي التولد وايضا تم برهن ان الاثر مخلص لكل مؤثر مقيد وك\لك قد تم التحقق من انه لاي مؤثر مقي

It was known that every left (?,?) -derivation is a Jordan left (?,?) – derivation on ?-prime rings but the converse need not be true. In this paper we give conditions to the converse to be true.

We present the concept of maps Γ- periodi2 on Γ -near-ring S. Our main goal is to research and explore the presence and mapping traits such as h Γ –hom anti-Γ –hom, Γ –α-derivations of Γ -periodi2 on Γ- near-rings.

In this paper, the computational complexity will be reduced using a revised version of the selected mapping (SLM) algorithm. Where a partial SLM is achieved to reduce the mathematical operations around 50%. Although the peak to average power ratio (PAPR) reduction gain has been slightly degraded, the dramatic reduction in the computational complexity is an outshining achievement. Matlab simulation is used to evaluate the results, where the PAPR result shows the capability of the proposed method.  

Let R be a 2-torision free prime ring and ?, ?? Aut(R). Furthermore, G: R×R?R is a symmetric generalized (?, ?)-Biderivation associated with a nonzero (?, ?)-Biderivation D. In this paper some certain identities are presented satisfying by the traces of G and D on an ideal of R which forces R to be commutative