Characterizations of core and dual core inverses in rings with involution. Issue 4 (3rd April 2018) - Li, Tingting; Chen, Jianlong
ABSTRACT: 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 p a = 0 and a n + p is invertible. As a consequence, a is an E P element if and only if there exists a Hermitian element (or a projection) p such that p a = a p = 0 and a n + 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 .