DOI: 10.28919/ejma.2022.2.10 Eur. J. Math. Appl. (2022)2:10 URL: http://ejma.euap.org © 2022 European Journal of Mathematics and Applications CYCLIC CODES AND PRIMITIVE IDEMPOTENTS IN THE FINITE CYCLIC GROUP ALGEBRAS AZHAR O. ALMALKI * AND AHMED A. KHAMMASH Abstract. We parallelly discuss the construction of cyclic linear codes as ideals in the finite cyclic group ring as well as zero-divisors therein. We also determine a complete set of primitive idempotents in the finite cyclic group ring over a field of characteristic p. Introduction Cyclic codes are among the most important types of codes in algebraic coding theory. They provide a substantial link between coding theory and various algebraic structures, and they are important for both theoretical and practical reasons; in fact, most existing linear codes in use are cyclic codes. The first connection between codes and group rings of finite groups appeared in the work of F.G. MacWilliams 1969 [4] in which cyclic codes were identified with ideals in the group algebras of cyclic groups (see also [5]), consequently, two sided ideals in a group algebra are named codes. Since then the algebraic structure of the group ring has been deeply involved in the study and constructions of codes. In particular properties of (central) primitive idempotents in the group algebra of finite groups over finite fields are heavily used in codes construction [6], [7]. On the other hand it is shown in [3] that cyclic codes are exactly zero-divisor codes in group rings of cyclic groups. Also Reed-Muller codes are extended cyclic codes and have been shown to be associated with the group ring of the elementary abelian 2-group [5]. In this paper, the zero divisor construction of cyclic codes is investigated in parallel with the ideal construction in the cyclic group ring. The main aim is to determine a complete set of primitive orthogonal idempotents of the group algebra of a cyclic group over a field of characteristic p (Theorem 3.1.1) and investigate the structure of the ideal codes (projective indecomposable modules) generated by primitive idempotents (Theorem 3.1.5). We also investigate the cyclic codes generated by those primitive idempotents as zero divisor type codes in the group ring of the cyclic group (Theorem 3.2.1). 1. Preliminaries Here we explain the concept of linear cyclic codes and how they are realized as ideals in the group rings of the cyclic groups as well as zero divisors therein. Department of mathematical sciences, Umm Al-Qura University, Saudi Arabia * Corresponding author E-mail addresses: azharobaidm@gmail.com, prof.khammash@gmail.com. Key words and phrases. cyclic codes; group rings; idempotents. Received 25/01/2022. 1