Nonlinear (skew-)centralizing mappings on unital algebras with nontrivial idempotents

Let \(\mathcal {R}\) $\mathcal{R}$ be a commutative ring with unity, and \(\mathcal {A}\) $\mathcal{A}$ be a unital \(\mathcal {R}\) $\mathcal{R}$ -algebra with a nontrivial idempotent. Under some mild conditions, we prove that every nonlinear centralizing mapping on \(\mathcal {A}\) $\mathcal{A}$ is proper. Nonlinear skew-centralizing mapping on \(\mathcal {A}\) $\mathcal{A}$ is also studied in this paper.