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 weighted core inverse in rings with involution
R is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse of an element in R by idempotents and units. For example, let a ∈ R and e ∈ R be an invertible Hermitian element, n > 1, then a is e-core invertible if and only if there exists an element (or an idempotent) p such that (ep) = ep, pa = 0 and a+p (or a(1−p)+p) is invertible. As a consequence, let e, f ∈ R be two invertible Hermitian elements, then a is weighted-EP with respect to (e, f) if and only if there exists an element (or an idempotent) p such that (ep) = ep, (fp) = fp, pa = ap = 0 and a + p (or a(1 − p) + p) is invertible. These results generalize and improve conclusions in [14].
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).