Jordan derivations and antiderivations of generalized matrix algebras (Q2852257)

Given a commutative ring \(\mathcal R\) with identity, a unital algebra \(A\) over \(\mathcal {R}\), and the center \(\mathcal {F}\) of \(A\), an \(\mathcal R\)-linear map \(\Theta_d:A\to A\) is called a derivation if it satisfies \(\Theta_d(ab) = \Theta_d(a) + a\Theta _d(b)\) for all \(a, b\in A\). An \(\mathcal R\)-linear map \(\Theta_{\text{Jord}}:A\to A\) is called a Jordan derivation if it satisfies \(\Theta_{\text{Jord}}(a^2) =\Theta_{\text{Jord}}(a)a + a\Theta_{\text{Jord}}(a) \) for all \(a\in A\). Clearly every derivation is a Jordan derivation but the converse is not necessarily true. Jordan derivations fail to be derivations are called proper Jordan derivation. An \(\mathcal R\)-linear map \(\Theta_{\text{Jord}}:A\to A\) is called an antiderivation if it satisfies \(\Theta_{\text{Jord}}(ab) =\Theta_{\text{Jord}}(b)a + b\Theta_{\text{Jord}}(a) \) for all \(a, b\in A\). In this paper, the authors study whether there are proper Jordan derivations for the generalized matrix algebra \(\mathcal{G}\) defined by Morita context \((A,B,_{A}M_B,_{B}N_A,\Phi_{MN}, \Psi_{NM})\).NEWLINENEWLINEExamples of generalized matrix algebras are given in Section 2. The characterization of Jordan derivations on \(\mathcal{G}\) is given in Proposition 3.2. In Corollary 3.3, the characterization of the Jordan derivation, when \(\mathcal{G}\) is a \(2\)-torsion free, is given. Example 3.5 exhibits a Jordan derivation, which is not a derivation. Proposition 3.10 shows that if one of the bilinear forms \(\Phi_{MN}\) and \(\Psi_{NM}\) is nondegenerate, antiderivations must be zero. Theorem 3.11 shows that \(\mathcal{G}\) is a \(2\)-torsion free generalized matrix algebra over \(\mathcal{R}\) and the bilinear pairings \(\Phi_{MN}\), \(\Psi_{NM}\) are both zero, then every Jordan derivation of \(\mathcal{G}\) can be expressed as the sum of a derivation and an antiderivation.