Additive and Product Properties of Drazin Inverses of Elements in a Ring

Abstract

We study the Drazin inverses of the sum and product of two elements in a ring. For Drazin invertible elements a and b such that a 2 b = aba and b 2 a = bab, it is shown that ab is Drazin invertible and that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible. Moreover, the formulae of (ab) D and (a + b) D are presented. Thus, a generalization of the main result of Zhuang, Chen et al. (Linear Multilinear Algebra 60 (2012) 903-910) is given. 1 The problem of Drazin inverse of the sum of two Drazin invertible elements was first considered by Drazin in his celebrated paper [7]. It was proved that (a + b) D = a D + b D under the condition that ab = ba = 0 in associative rings. It is well known that the product ab of two commutative Drazin invertible elements a, b is Drazin invertible and (ab) D = a D b D = b D a D in a ring. In 2011, Wei and Deng [12] considered the relations between the Drazin inverses of A + B and 1 + A D B for two commutative complex matrices A and B. For two commutative Drazin invertible elements a, b ∈ R,Zhuang, Chen et al. [16] proved that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible.Moreover, the representation of (a + b) D was obtained. More results on Drazin inverse can be found in [1][2][3][4][5][6]8,[11][12][13][14][15][16].

Additive properties of central Drazin invertibility of elements in a ring

“…Under the conditions P 2 Q = PQP and Q 2 P = QPQ, Liu et al [14] characterized the relation between Drazin inverses of P + Q and I + P D Q for complex matrices P and Q by using the methods of splitting complex matrices into blocks. The results in [22] and [14] were extended to the condition of a 2 b = aba and b 2 a = bab in an associative ring by Zhu and Chen [21] in 2017.…”

Block representations for the g-Drazin inverse in Banach algebras

“…The g-Drazin inverse of the sum of two elements in a Banach algebra has been studied by many authors, e.g. [3,6,8,10,11] and [12]. In [9, Theorem 2.1], Yang and Liu gave the representation of the Drazin inverse of two complex matrices P and Q such that PQP = 0, PQ 2 = 0, which recovered the case PQ = 0 studied by Hartwig et al In [7], Ljubisavljevic and Cvetkovic-Ilic derived an expression of (P + Q) D under a weaker condition PQP 2 = 0, PQPQ = 0, PQ 2 P = 0 and PQ 3 = 0.…”

On forward-order law for core inverse in rings

“…In 2012, Mosic and Djordjevic [10] extended the reverse-order law for the group inverse in Hilbert space to ring. In 2017, Zhu and Chen [19] bestowed the forward-order law for the Drazin inverse in a ring. Zhu ([21] and [22]) conferred several results on additive properties, reverse-order law and forward-order law.…”