On <em>n</em>-Derivations and <em>n</em>-Homomorphisms in Perfect Lie Superalgebras

Let n ≥ 2 be a fixed integer. The main aim of this paper is to investigate the properties of n-derivations within the framework of perfect Lie superalgebras over a commutative ring R. The main result shows that, if the base ring contains 1/n−1, and L is a perfect Lie superalgebra with a center equal to zero, then any n-derivation of L is necessarily a derivation. Additionally, every n-derivation of the derivation algebra Der(L) is an inner derivation. Finally, extend the concept of n-homomorphisms to mappings between Lie superalgebras L and L′, and prove that under specific assumptions, homomorphisms, anti-homomorphisms, and their combinations are all n-homomorphisms.