m -weak group inverses in a ring with involution
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m-weak group inverses in a ring with involution Yukun Zhou1 · Jianlong Chen1 · Mengmeng Zhou1 Received: 12 April 2020 / Accepted: 31 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In a unitary ring with involution, we prove that each element has at most one weak group inverse if and only if each idempotent element has a unique weak group inverse. Furthermore, we define the m-weak group inverse and show some properties of m-weak group inverse. Keywords Weak group inverse · m-Weak group inverse · Pseudo core inverse Mathematics Subject Classification 15A09 · 16W10
1 Introduction In 1958, Drazin [4] introduced the definition of the pseudo inverse in rings and semigroups, which is called the Drazin inverse later. It’s well known that an element is Drazin invertible if and only if this element is strongly π-regular [16] and [17] give some applications of the Drazin inverse. In 2014, Manjunatha Prasad et al. [12] introduced the core-EP inverse of a complex matrix, which is the generalization of the core inverse [1]. In 2017, Gao et al. [7] generalized the core-EP inverse to rings with involution, where an involution a → a ∗ is an anti-isomorphism such that (a + b)∗ = a ∗ + b∗ , (ab)∗ = b∗ a ∗ , (a ∗ )∗ = a for any a, b ∈ R. For more details of the core-EP inverse, readers can see [2,5,15,21]. In the rest of this paper, we restrict R is a unitary ring with involution. In 2018, Wang et al. [19] defined the weak group inverse of a complex matrix, which is different from other generalized inverses, such as the Moore-Penrose inverse [14], the DMP inverse [11]. Then, Zhou et al. [22] generalized the weak group inverses to proper ∗-rings and proved that each element in a proper ∗-ring has at most one weak group inverse. They gave an example to show that the weak group inverse of an element may not be unique in R. In addition, they proved that an element in a proper ∗-ring is weak group invertible if and only if this element has group-EP decomposition. For more details of the weak group inverse, readers can see [6,19].
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Jianlong Chen [email protected] Yukun Zhou [email protected] Mengmeng Zhou [email protected]
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School of Mathematics, Southeast University, Nanjing 210096, China 0123456789().: V,-vol
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Y. Zhou et al.
However, the condition that R is a proper ∗-ring is a sufficient but not necessary condition under which each element has at most one weak group inverse (see Example 3.8). This motivates us to study sufficient and necessary conditions under which each element has at most one weak group inverse. In addition, it’s natural to consider the conditions under which an element is weak group invertible. Motivated by the idea of [10] and [23], we define the m-weak group inverse in R. Then we give an equivalent definition of the m-weak group inverse. In addition, we consider the relation between the m-weak group inverse and the (m + 1)-weak group inverse. Some equivalent characterizations are given when the m-weak group inverse is equal to the Drazin inverse. The rest of this pa
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