Generators of the Augmentation Ideal in a Group Ring R[G]

Abstract

This research paper provides an important information about the special type of ideal, that is augmentation ideal. We use the concept of augmentation ideal in algebraic structure group ring R [ G ]. Let us suppose that there be a homomorphism f such that,        : here homomorphism f is known as augmentation map. But the kernel of f this means ker f is termed as augmentation ideal. Thus it is ovious that it is a special type of ideal. This paper also describes the properties of augmentation ideal as it is a left as well as a right ideal in R [ G ]. This ideal is generated by the difference of group elements G . When we use identity element of group G then it will be generated by ( g – e ), but group G is based on multiplication operation so the given augmentation ideal is generated by ( g – 1 g ). Since, group ring algebraic structure R [ G ] is a ring so it must have ideals. But here we have discussed upon augmentation ideal of R [ G ] which is other then its normal ideals. We have also proved some theorems as well as lemmas are based on the concept of augmentation ideal and generators of the augmentation