Central elements of the Jennings basis and certain Morita invariants

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th (right) socle $ZS^n(A) := Z(A) \cap \operatorname{Soc}^n(A)$ of a finite-dimensional algebra $A$ is a Morita invariant; This is a generalization of important Morita invariants --- the center $Z(A)$ and the Reynolds ideal $ZS^1(A)$. As an example, we also studied $ZS^n(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent set of $ZS^n(FG)$. In fact, such elements form a basis of $ZS^n(FG)$ for every integer $1 \le n \le p$ if $G$ is powerful. As a corollary we have $\operatorname{Soc}^p(FG) \subseteq Z(FG)$ if $G$ is powerful.