New characterizations for core inverses in rings with involution

doi:10.1007/s11464-016-0591-2

Front. Math. China

2017, Vol. 12

Issue (1) : 231-246 https://doi.org/10.1007/s11464-016-0591-2

RESEARCH ARTICLE

New characterizations for core inverses in rings with involution

Sanzhang XU,Jianlong CHEN(

),Xiaoxiang ZHANG

Department of Mathematics, Southeast University, Nanjing 210096, China

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Abstract

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakíc, N.Č. Diňcíc and D. S. Djordjevíc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R^# with a^# = b if and only if (ab)∗ = ab, ba²= a, and ab²= b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

Keywords Core inverse dual core inverse group inverse {1,3}-inverse {1,4}-inverse

Corresponding Author(s): Jianlong CHEN

Issue Date: 17 November 2016

Cite this article:

Sanzhang XU,Jianlong CHEN,Xiaoxiang ZHANG. New characterizations for core inverses in rings with involution[J]. Front. Math. China, 2017, 12(1): 231-246.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0591-2
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I1/231

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