  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.
