ON NIL-SYMMETRIC RINGS AND MODULES SKEWED BY RING ENDOMORPHISM - Mustafa, Ibrahim; Ahmed, Chnar Abdulkareem (Science journal of University of Zakho (Online), 2025)
The symmetric property plays an important role in non-commutative ring theory and module theory.  In this paper, we study the symmetric property with one element of the ring  and two nilpotent elements of  skewed by ring endomorphism  on rings, introducing the concept of a right - -symmetric ring and extend the concept of right - -symmetric rings to modules by introducing another concept called the right - -symmetric module which is a generalization of -symmetric modules. According to this, we examine the characterization of a right - -symmetric ring and a right  - -symmetric module and their related properties including ring and explore their connections to other classes of rings and modules.   Furthermore, we investigate the concept of - -symmetric on some ring extensions and localizations like  Dorroh extension, Jordan extension and module localizations like