A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if for all [Formula: see text], [Formula: see text] is nilpotent, if and only if for all [Formula: see text], [Formula: see text] is strongly nil-clean, if and only if every element in [Formula: see text] is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if [Formula: see text] is tripotent and [Formula: see text] is nil, if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] is a Boolean ring and [Formula: see text] is nil; [Formula: see text] is a Yaqub ring and [Formula: see text] is nil. Strongly 2-nil-clean group algebras are investigated as well.
We introduce and study a new class of Drazin inverses. An element a in a ring has Hirano inverse b if a 2 −ab ∈ N (R), ab = ba and b = bab. Every Hirao inverse of an element is its Drazin inverse. We derive several characterizations for this generalized inverse. An element a ∈ R has Hirano inverse if and only if a 2 has strongly Drazin inverse, if and only if a − a 3 ∈ N (R). If 1 2 ∈ R, we prove that a ∈ R has Hirano inverse if and only if there exists p 3 = p ∈ comm 2 (a) such that a − p ∈ N (R), if and only if there exist two idempotents e, f ∈ comm 2 (a) such that a + e − f ∈ N (R). Clines formula and additive results for this generalized inverse are thereby obtained.2010 Mathematics Subject Classification. 15A09, 16E50, 15A23.
Let A be a Banach algebra. An element a ∈ A has generalized Hirano inverse if there exists b ∈ A such that b = bab, ab = ba, a 2 − ab ∈ A qnil .We prove that a ∈ A has generalized Hirano inverse if and only if a has g-Drazin inverse and a− a 3 ∈ A qnil , if and only if there exists p 3 = p ∈ comm(a) such that a − p ∈ A qnil . The Cline's formula for generalized Hirano inverses are thereby obtained. Let a, b ∈ A have generalized Hirano inverse. If a 2 b = aba and b 2 a = bab, we prove that a + b has generalized Hirano inverse if and only if 1 + a d b has generalized Hirano inverse. Hirano inverses of operator matrices over Banach spaces are also studied.2010 Mathematics Subject Classification. 15A09, 32A65, 16E50.
Let R be an associative ring with an identity and suppose that a, b, c, d ∈ R satisfy bdb = bac, dbd = acd.If ac has generalized Drazin ( respectively, pseudo Drazin, Drazin) inverse, we prove that bd has generalized Drazin (respectively, pseudo Drazin, Drazin) inverse. In particular, as applications, we obtain new common spectral properties of bounded linear operators over Banach spaces.2010 Mathematics Subject Classification. 15A09, 47A11, 47A53, 16U99.
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