Eksistensi Invers Moore Penrose Diperumum Elemen Normal Diperumum pada Ring dengan Involusi

Abstract

The Moore Penrose inverse of normal element in ring with involution  have been discussed by several researchers. By generalizing  concept of Moore Penrose inverse to the generalized Moore Penrose inverse, the element properties of the generalized Moore Penrose inverse of normal elements have also been obtained. In addition to generalizing the concept of Moore Penrose inverse, the definition of the normal element has also been generalized by generalizing the power of 1 to  n ∈ N. It is found that  intersection between set of  generalized normal element  and set of generalized  Moore Penrose inverse element is not empty. This indicates that both of them have common properties, so this paper aims is to build the necessary and sufficient conditions for a generalized normal element to have a generalized Moore Penrose inverse using these properties. The method used is  to look for the similarity of properties possessed by a generalized normal element and  element that has generalized Moore Penrose inverse. The next step is to use the involution properties to obtain the final result. The approach taken is not only through the generalized Moore Penrose inverse, but also  group inverse.