566 research outputs found
Characterizing Jordan derivations of matrix rings through zero products
Let \Mn be the ring of all matrices over a unital ring
, let be a 2-torsion free unital \Mn-bimodule and
let D:\Mn\rightarrow \mathcal{M} be an additive map. We prove that if
D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0 whenever \A,\B\in \Mn are such that
\A\B=\B\A=0, then D(\A)=\delta(\A)+\A D(\textbf{1}), where
\delta:\Mn\rightarrow \mathcal{M} is a derivation and lies in
the centre of . It is also shown that is a generalized
derivation if and only if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A
D(\textbf{1})\B-\B D(\textbf{1})\A=0 whenever \A\B=\B\A=0. We apply this
results to provide that any (generalized) Jordan derivation from \Mn into a
2-torsion free \Mn-bimodule (not necessarily unital) is a (generalized)
derivation. Also, we show that if \varphi:\Mn\rightarrow \Mn is an additive
map satisfying \varphi(\A \B+\B \A)=\A\varphi(\B)+\varphi(\B)\A \quad (\A,\B
\in \Mn), then \varphi(\A)=\A\varphi(\textbf{1}) for all \A\in \Mn, where
lies in the centre of \Mn. By applying this result we
obtain that every Jordan derivation of the trivial extension of \Mn by \Mn
is a derivation.Comment: To appear in Mathematica Slovac
2n-Weak module amenability of semigroup algebras
Let be an inverse semigroup with the set of idempotents . We prove
that the semigroup algebra is always -weakly module amenable
as an -module, for any , where acts on
trivially from the left and by multiplication from the right.Comment: arXiv admin note: text overlap with arXiv:1207.4514 by other author
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