Every module has an injective hull, a quasi-injective hull, and a quasi-continuous hull. However, there exists a uniform nonsingular cyclic module over a noncommutative ring which has no continuous hull. (Recall that the continuous hull of a module
M
M
is the smallest continuous extension of
M
M
in a fixed injective hull
E
(
M
)
E(M)
of
M
M
.) When the base ring is commutative, every uniform cyclic module over a commutative ring has a continuous hull. Further, every nonsingular cyclic module over a commutative ring has a continuous hull. Motivated by the example and these results on continuous hulls, it is interesting to study whether
R
R
R_R
has a continuous hull for any ring
R
R
. We show that for any integer
n
n
such that
n
>
1
n>1
, if
R
R
is the
n
×
n
n\times n
matrix ring over any given ring or the
n
×
n
n\times n
upper triangular matrix ring over any given ring, then
R
R
R_R
has a continuous hull and such a continuous hull is explicitly described. Moreover, if
R
R
is an abelian regular ring, it is shown that every nonsingular cyclic right
R
R
-module has a continuous hull. As an application, we show that
R
R
R_R
has a continuous hull when
R
R
is an abelian regular ring. In this case,
R
B
(
Q
(
R
)
)
R
R\mathcal {B}(Q(R))_R
(where
R
B
(
Q
(
R
)
)
R\mathcal {B}(Q(R))
is the idempotent closure of
R
R
) is the continuous hull of
R
R
R_R
. As a byproduct, we show that when
R
R
is an abelian regular ring,
R
B
(
Q
(
R
)
)
R\mathcal {B}(Q(R))
is the smallest continuous regular ring of quotients of
R
R
. We discuss the condition (
⋆
\star
) of a commutative ring
R
R
, which is precisely that the classical ring of quotients of every uniform factor ring of
R
R
is self-injective. Moreover, by using the condition (
⋆
\star
), we provide a detailed proof that every module over a commutative noetherian ring has a continuous hull. Various examples which illustrate our results are provided.