Lie (Jordan) Centralizer on Graded Rings

In this paper, we provide the necessary and sufficient conditions for a Lie centralizer to be proper and a Jordan centralizer to be a centralizer on graded rings. Since every Lie centralizer naturally induces an anti-symmetric mapping and every Jordan centralizer naturally induces a symmetric mapping, our results provide the underlying graded structures reflected in these mappings. As applications, we recover known results for trivial extension algebras and triangular algebras, and additionally characterize Lie (Jordan) centralizers on exterior algebras, whose operations are inherently anti-symmetric.