| Abstrakt: |
Let $ \mathbb {F} $ F be a field, $ n\geq ~3 $ n ≥ 3 and let $ M_n $ M n be the ring of all $ n\times n $ n × n matrices with entries in $ \mathbb {F}. $ F. For a given subset H of $ M_n, $ M n , consider $ \bar {H}=H\cup \mathbb {V} $ H ¯ = H ∪ V with $ \mathbb {V}=\{\alpha E_{ii}\mid \alpha \in \mathbb {F}\ \mbox {and}\ E_{ii}\ \mbox {is a matrix unit for each}\ i=1,\ldots,n\}. $ V = { α E ii ∣ α ∈ F and E ii is a matrix unit for each i = 1 , ... , n }. In this paper, under a mild technical assumption on $ \mathbb {F}, $ F , we describe additive maps $ G:M_n\to M_n $ G : M n → M n satisfying $ [G(X),X]=G(X)X-XG(X)=0 $ [ G (X) , X ] = G (X) X − XG (X) = 0 for all $ X\in \bar {H} $ X ∈ H ¯ in the following settings: $ H=\mathbb {E}=\{A\in M_n\mid A\ \mbox {is idempotent}\}; $ H = E = { A ∈ M n ∣ A is idempotent } ; $ H=S=\{A\in M_n\mid A\ \mbox {is algebraic of degree}\ 2\}; $ H = S = { A ∈ M n ∣ A is algebraic of degree 2 } ; $ H=\mathbb {P}=\{X\in M_n(\mathbb {F})\mid \mbox {tr}(X)\in \mathbb {L}\} $ H = P = { X ∈ M n (F) ∣ tr (X) ∈ L } where $ \mathbb {L} $ L is the prime field of $ \mathbb {F}. $ F. These maps are so-called commuting on $ \bar {H}. $ H ¯. Our findings will allow us to conclude that for H = S the map G has the so-called standard form. Moreover, we will show that G is commuting on $ \overline {\mathbb {E}} $ E ¯ if and only if G is commuting on $ \overline {\mathbb {P}}. $ P ¯. Our investigation will lead us to consider functional identities on $ \mathbb {F} $ F of the form $ H_{1}(r)s+H_{2}(s)r=0 $ H 1 (r) s + H 2 (s) r = 0 with r, s belonging to a 2-point subset of $ \mathbb {F}. $ F. At the end, we will discuss about the existence of non-standard additive commuting maps on $ \overline {\mathbb {E}} $ E ¯ which are not commuting on the set of rank 1 matrices and vice versa. [ABSTRACT FROM AUTHOR] |
