Let \(\mathcal {A}\) be a unital algebra and \(\mathcal {M}\) be a unital \(\mathcal {A}\)-bimodule. We characterize the linear mappings \(\delta \) and \(\tau \) from \(\mathcal {A}\) into \(\mathcal {M}\), satisfying \(\delta (A)B+A\tau (B)=0\) for every \(A,B \in \mathcal {A}\) with \(AB=0\) when \(\mathcal {A}\) contains a separating ideal \(\mathcal {T}\) of \(\mathcal {M}\), which is in the algebra generated by all idempotents in \(\mathcal {A}\). We apply the result to \(\mathcal {P}\)-subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that \(\mathcal {A}\) is a unital Banach algebra and \(\mathcal {M}\) is a unital Banach \(\mathcal {A}\)-bimodule, we give a complete description of linear mappings \(\delta \) and \(\tau \) from \(\mathcal A\) into \(\mathcal M\), satisfying \(\delta (A)B+A\tau (B)=0\) for every \(A,B\in \mathcal {A}\) with \(AB=I\).

假设\（\ mathcal {A} \）为单位代数，\（\ mathcal {M} \）为单位\（\ mathcal {A} \）- bimodule。我们将线性映射\（\ delta \）和\（\ tau \）从\（\ mathcal {A} \）到\（\ mathcal {M} \）进行表征，满足\（\ delta（A）B + A \ tau蛋白（B）= 0 \）每\（A，B \在\ mathcal {A} \）与\（AB = 0 \）时\（\ mathcal {A} \）含有分离理想\（\ mathcal【T} \）的\（\ mathcal {M} \） ，这是在由所有幂等所产生的代数\（\ mathcal {A} \） 。我们将结果应用于\（\ mathcal {P} \）-子空间格代数，完全分布式可交换子空间格代数和unit元标准算子代数。此外，假设\（\ mathcal {A} \）是一个单位Banach代数，\（\ mathcal {M} \）是一个单位Banach \（\ mathcal {A} \）- bimodule，我们给出完整的描述线性映射\（\ delta \）和\（\ tau \）从\（\ mathcal A \）到\（\ mathcal M \），满足\（\ delta（A）B + A \ tau（B）= 0 \）与\（AB = I \）中的每个\（A，B \在\ mathcal {A} \）中。