Characterizations of core and dual core inverses in rings with involution
Tingting Li, Jianlong Chen · Linear and Multilinear Algebra · 2017
Let R be a unital ring with involution. We give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let a∈R and n⩾1. Then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa=0 and an+p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa=ap=0 and an+p is invertible. We also get a new characterization for both core invertibility and dual core invertibility of a regular element by units, and their expressions are shown. In particular, we prove that for n⩾2, a is both Moore–Penrose invertible and group invertible if and only if (a∗)n is invertible along a.