Abstract
Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent.
Highlights
In this paper, G is referred as a finite abelian group
In [7], we have fully identify all idempotents in every F2Cn algebraically, where Cn is a cyclic group, by using our concept of basis idempotents
We introduce a way to partition the support of eL into the supports of some idempotents which we defined as basis idempotents
Summary
G is referred as a finite abelian group. Let Fq be a finite field of q elements. In 2006, Ted Hurley and Paul Hurley introduced a modern approach to study FqG codes, which are submodules of the FqG [4]. This inspires us to study the potential of having idempotent generators for zero-divisor codes in determining their properties. In [7], we have fully identify all idempotents in every F2Cn algebraically, where Cn is a cyclic group, by using our concept of basis idempotents. We prove that the support of all non-zero generated idempotents in F2G can be obtained by partitioning the support of the largest finite sum of generated idempotents and we identify all the idempotents in F2G
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
