LAUSR

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