On the algebraic closure in rings
The "algebraic closure" of a subset K C A of a ring is an algebraic analogue of topological closure.
Characterizations of core and dual core inverses in rings with involution
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 and . Then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that and is invertible. As a consequence, a is an element if and only if there exists a Hermitian element (or a projection) p such that and 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 , a is both Moore–Penrose invertible and group invertible if and only if is invertible along a.
Closure of invertible operators on a Hilbert space
Although most of the characterizations of the closure of the invert- ible operators on a separable Hubert space do not extend to a nonseparable Hubert space, this note gives a characterization for an arbitrary Hubert space that generalizes the separable case in a natural way. The new concept of essential nullity, which facilitates this characterization, should find other applications.