College of Computer Science and Mathematics

Discussion of the master’s thesis in the College of Computer Science and Mathematics -Department of

Mathematics Sciences entitled:

“ 5- Potent Nil Clean Rings“

It was discussed in the discussion room at the Faculty of Computer Science and Mathematics at the University of Mosul on Thursday , 28 -8-2025.

Master’s thesis by Student  Ali Khalid Abbo Matroud

 under the supervision of Prof. Dr. Nazar Hamdoon Shuker

As a generalization of tripotent rings, the idea of five potent rings was introduced. These are rings where every element satisfies the equation . This gives a new way to study rings that have special power properties. Some basic properties of these rings were studied, like how their elements behave with addition and multiplication, and how they relate to special types of elements like idempotents and units.

A new type of ring was also defined, called the five-potent nil clean ring, or 5-PNC ring for short. In these rings, every element can be written as the sum of a five potent element and a nilpotent (an element that becomes zero when raised to some power). This is a generalization of the usual nil clean rings.

The relationships between 5-PNC rings and other rings like nil clean rings, strongly 2-nil clean rings, and tripotent rings were also studied. The similarities and differences between them were explained, and some results were given about their structure.

These new ideas may help researchers understand rings better and may lead to new studies in algebra and other areas of mathematics.

The scientific committee included the following members:

• Dr. Akrm Salim mohammed – Chairman
• Dr. Nada Yasen Qasim – Member.
• Prof. Dr. Ekram Abdulqader Saleh – Member.
• Dr. Nazar Hamdoon Shuker – Member and supervisor.