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From Morita theoretic viewpoint, computing Morita invariants is important. Weprove 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 aMorita invariant; This is a generalization of important Morita invariants ---the center $Z(A)$ and the Reynolds ideal $ZS^1(A)$. As an example, we alsostudied $ZS^n(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over afield $F$ of positive characteristic $p$. Such an algebra has a basis along thesocle filtration, known as the Jennings basis. We prove certain elements of theJennings 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.

Central elements of the Jennings basis and certain Morita invariants