International Mathematical Forum, 4, 2009, no. 47, 2317 - 2325 s-Weakly Regularity of Group Algebras with GAP Alper Odaba¸ s Eski¸ sehir Osmangazi University, Art and Science Faculty Department of Mathematics and Computer Sciences TR-03200, Eski¸ sehir, Turkey aodabas@ogu.edu.tr Abstract In this paper we describe s-weakly regular group algebras with GAP. We present a table of the s-weakly regular group algebras on groups of order at most 30. Mathematics Subject Classification: 16D40, 16S34, 20C05 Keywords: Group algebra, n-regular rings, s-weakly regular ring, GAP 1 Introduction GAP (Groups, Algorithms, Programming) [12] system for computational discrete mathematics has a number of novel features. The system is central to the organization of the library which is the main part of the GAP system. Unlike simpler object-oriented systems, GAP allows method selection based on the types of all arguments and on certain aspects of the relationship between the arguments. [5] Although GAP has powerful database it does not contain a special function for regular ring. (and regular group algebras) Such a function was written by the author in [13]. In this study using this function with the theoretical knowledge it is determined for a given group algebra that the group algebra is s-weakly regular or not. As it is well-known, a ring is said to be Neumann regular if the equation axa = a has a solution x ∈ R for any a ∈ R. There are several generalizations of regularity, for instance, n-weakly regular [6] and n-regular rings [1]. Definition 1.1. A ring R is s-weakly regular if for each a ∈ R, a ∈ aRa 2 . If for any a 1 , ..., a n ∈ R there exist x 1 , ..., x n ∈ R with R(a 1 - a 1 x 1 a 1 )R...R(a n - a n x n a n )R =0