Lie superderivations on unital algebras with idempotents

Let U be an associative unital algebra containing a non-trivial idempotent e. We consider U as a superalgebra whose Z2-grading is induced by e. This paper aims to describe Lie superderivations of U. In particular, we characterize the general form of Lie superderivations of U and apply it to present the necessary and sufficient conditions for a Lie superderivation on U to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their Z2-grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras.