Let R be an associative ring, not necessarily commutative and not necessarily having unity. Recall an element x in R is called quasi-regular if and only if solutions y and z exist for the equations x+ y - x*y = 0 and x + z - z*x = 0. In this case y=z, and the unique element y is called the quasi-inverse for x. It is well known that J(R), the Jacobson radical of R, is the unique largest ideal in R consisting entirely of quasi-regular elements. In this paper, we explore the implications of the case x is its own quasi-inverse. We call such elements self-quasi-regular. We determine some properties of sq(R), the set of all self-quasi-regular elements, for a general ring, and also compare this set to J(R). Then, we completely characterize the set sq(R) for all homomorphic images of Z, the integers, including the cardinality and membership of the set sq(Zn) for each choice of n.

Author Bio

I wrote this paper during an independent study class in Abstract Algebra (specifically ring theory). I graduated in the Spring of 2006 from Georgia College & State University as the Phi Kappa Phi Honor graduate with an overall 4.0 GPA. I am currently a graduate student at Georgia Institute of Technology. I plan on using my mathematics ability to model financial markets. I really enjoy sports (especially baseball, football, basketball, golf, tennis, etc.). I also enjoy reading, playing jazz piano, and listening to a variety of different music.