  Matlis showed that the injective hull of a simple module over a commutative
Noetherian ring is Artinian. Many non-commutative Noetherian rings whose
injective hulls of simple modules are locally Artinian have been extensively
studied recently. This property had been denoted by property $(\diamond)$. In
this paper we investigate, which non-Noetherian semiprimary commutative
quasi-local rings $(R, m)$ satisfy property $(\diamond)$. For quasi-local rings
$(R,m)$ with $m^3=0$, we prove a characterisation of this property in terms of
the dual space of $Soc(R)$. Furthermore, we show that $(R,m)$ satisfies
$(\diamond)$ if and only if its associated graded ring $gr(R)$ does.
  Given a field $F$ and vector spaces $V$ and $W$ and a symmetric bilinear map
$\beta:V\times V\rightarrow W$ we consider commutative quasi-local rings of the
form $F\times V \times W$, whose product is given by $(\lambda_1,
v_1,w_1)(\lambda_2,v_2,w_2) = (\lambda_1\lambda_2, \lambda_1v_2+\lambda_2v_1,
\lambda_1w_2+\lambda_2w_1+\beta(v_1,v_2))$ in order to build new examples and
to illustrate our theory. In particular we prove that any quasi-local
commutative ring with radical cube-zero does not satisfy $(\diamond)$ if and
only if it has a factor, whose associated graded ring is of the form $F\times V
\times F$ with $V$ infinite dimensional and $\beta$ non-degenerated.
