Let $ \mathcal{D} $ be a division ring with char$ (\mathcal{D}) = 2 $. Let $ \mathcal{R} = M_n(\mathcal{D}) $ be the ring of all $ n\times n $ matrices over $ \mathcal{D} $, where $ n\geq 2 $. Let $ f, g:\mathcal{R}\rightarrow \mathcal{R} $ be two additive maps such that
$ f(A)+Ag(A^{-1})A = 0 $
for all invertible $ A\in\mathcal{R} $. If $ |\mathcal{D}|\neq 2, 4, 8 $, then
$ f(A) = AQ+\delta(A)\quad\text{and}\quad g(A) = QA+\delta(A) $
for all invertible $ A\in\mathcal{R} $, where $ A\in \mathcal{R} $ and $ \delta $ is a derivation of $ \mathcal{R} $. This result affirmatively answers a question posed by Argac et al. under a mild condition.
Citation: Yingyu Luo, Qian Chen. Identities with inverses on matrix rings over a division ring of characteristic two[J]. AIMS Mathematics, 2026, 11(3): 5283-5298. doi: 10.3934/math.2026217
Let $ \mathcal{D} $ be a division ring with char$ (\mathcal{D}) = 2 $. Let $ \mathcal{R} = M_n(\mathcal{D}) $ be the ring of all $ n\times n $ matrices over $ \mathcal{D} $, where $ n\geq 2 $. Let $ f, g:\mathcal{R}\rightarrow \mathcal{R} $ be two additive maps such that
$ f(A)+Ag(A^{-1})A = 0 $
for all invertible $ A\in\mathcal{R} $. If $ |\mathcal{D}|\neq 2, 4, 8 $, then
$ f(A) = AQ+\delta(A)\quad\text{and}\quad g(A) = QA+\delta(A) $
for all invertible $ A\in\mathcal{R} $, where $ A\in \mathcal{R} $ and $ \delta $ is a derivation of $ \mathcal{R} $. This result affirmatively answers a question posed by Argac et al. under a mild condition.
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Yingyu Luo, Qian Chen. Identities with inverses on matrix rings over a division ring of characteristic two[J]. AIMS Mathematics, 2026, 11(3): 5283-5298. doi: 10.3934/math.2026217
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