Generalized Inverse Of Sums And Products Of Elements In Rings

Abstract:The generalized inverse theory plays a important role in many fields, and a number of scholars extensively studied generalized inverses of complex matrices, bounded linear operators on Banach spaces, Banach algebras, C*-algebras, rings, semigroups. However, there are still some problems need to be researched. This paper mainly discusses generalized inverses of sums and products of some special elements in rings, and it consists of two parts.The first part mainly discusses around some properties of idempotents and generalized projectors. On the one hand, some necessary and sufficient conditions for the idempotency of the commutator C=pq-qp and the anticommutator D=pq+qp are obtained. Furthermore, we show that the difference of two idempotents is Drazin invertible when the square of com-mutator is equal to 0, which generalize the related results on a Hilbert space obtained by C. Y. Deng. On the other hand, the EP property of sums and products of generalized projectors and hyper-generalized projectors are also investigated in rings with involution, which generalize the results on bounded linear operators on a Hilbert space got by C. Y. Deng to an arbitrary ring with involution.The second part mainly discusses the group invertibility or invertibility of sums and products of group invertible elements in an algebra A over any field F. Firstly, we consider the invertibility of the combination of group invertible elements when some conditions are satisfied. Secondly, let t=c1t1+c2t2+C3t3, where t1,t2,t3 are three mutually commuting tripotent elements in A and c1,c2,c3 are numbers in F. The group invertibility of t is investigated and expressed. In addition, let t1,t2 be two tripotent elements in A and c1,c2,c3,c4 four numbers in F. We mainly study the nonsingularity of c1t1+c2t2-c3t1t2+c4t1t2t1 and c1t1+c2t2-c3t1t2+C1t2t1t2. Furthermore, some formulae for the inverse of them are given under some conditions. The above results respectively generalize related results of X. J. Liu and J. Benitez etc on complex matrices.