Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/1656 2022 2022 Faculty of Science, Department of Mathematics, Idlib University, Syria. Nader Taffach The problem of the existence and construction of a resolution of singularities is one of the central questions of algebraic geometry. In this paper, we study this problem in connecting with the quotients for . It is known that the action of  on its Lie algebra is corresponding to the action of  on . As a result of this action, it will be an invariant ring, which determines the quotients for . This paper is devoted to studying the singularity of these quotients. We write this singularity as a matrix with interesting features such as, for example, its quadratic is a zero matrix and its rank is less than or equal to 1. Therefore, in this paper, we reduce the studying of the singularity of the quotients of , which is a hard problem, to the studying of a matrix of invariants which is an easy problem. 2023 2023 24 30 10.54216/GJMSA.040202 https://www.americaspg.com/articleinfo/33/show/1656