Properties Of Rings Related To Nilpotent And Idempotent

The ring structure is one of the most basic structures in algebraic structures,and the properties of the elements in the ring are inseparable from the ring structure.In recent years,researchers focus on Nilpotent,idempotent,and ring structures.And this paper crosses the ring structure related to the nilpotent property from the perspective of the nilpotent element and gives the structural characteristics of several rings related to nilpotent and idempotent.The nilpotent property of elements is one of the most fundamental elemental properties in the field of algebra and has an extremely important position in the field of algebra.In recent years,with the deepening of the study of algebraic semimonographic structures,the nilpotent structure has received more and more attention.Among them,the nilpotent element is of great significance,especially the nilpotent matrix.Because the sum of the identity matrix and the nilpotent matrix constitutes the unipotent matrix.Unipotent,unipotent properties are closely associated with nilpotent properties.When the linear representation of dimension 11,the first thing to do in this paper is whether some exception of the free group of primitive elements is unipotent constitutes a unipotent group.An affirmative answer was given.The elements in a unipotent group are naturally the sum of the identity element and the nilpotent element.Generalizing this structure is the structural problem of a ring in which each element be constituted as the sum of nilpotent and idempotent elements.Further down is the case expressed as the sum of k-idempotent and nilpotent elements.Therefore,the structure of such rings is considered next.Among them,the characteristics of the nilpotent element once again played a crucial role.We conclude that the ring can be represented as the direct sum of several characteristically fixed rings.A commutative ring that does not contain a nilpotent element must be regarded as the direct sum of subrings in a set of domains.Therefore,reversibility rejects nilpotent.So,is the generalized inverse compatible with nilpotent? Therefore,the final study of this paper is the generalized inverse of matrices.We give a more detailed description of the generalized inverse and find a new type of generalized inverse between the two types of the generalized inverse.