A characterization is given of those Noetherian rings
R
R
such that whenever
R
R
is ideally closed (
≡
\equiv
cyclically pure) in an extension algebra
S
S
, then
R
R
is pure in
S
S
. In fact,
R
R
has this property if and only if the completion
(
A
,
m
)
(A,m)
of each local ring of
R
R
at a maximal ideal has the following two equivalent properties: (i) For each integer
N
>
0
N > 0
there is an
m
m
-primary irreducible ideal
I
N
⊂
m
N
{I_N} \subset {m^N}
. (ii) Either
dim
A
=
0
\dim \;A = 0
and
A
A
is Gorenstein or else depth
A
⩾
1
A \geqslant 1
and there is no
P
∈
Ass
(
A
)
P \in {\operatorname {Ass}}(A)
such that
dim
(
A
/
P
)
=
1
\dim (A/P) = 1
and
(
A
/
P
)
⊕
(
A
/
P
)
(A/P) \oplus (A/P)
is embeddable in
A
A
. It is then shown that if
R
R
is a locally excellent Noetherian ring such that either
R
R
is reduced (or, more generally, such that
R
R
is generically Gorenstein), or such that Ass(
R
R
) contains no primes of coheight
⩽
1
\leqslant 1
in a maximal ideal, and
R
R
is ideally closed in
S
S
, then
R
R
is pure in
S
S
. Matlis duality and the theory of canonical modules are utilized. Module-theoretic analogues of condition (i) above are, of necessity, also analyzed. Numerous related questions are studied. In the non-Noetherian case, an example is given of a ring extension
R
→
S
R \to S
such that
R
R
is pure in
S
S
but
R
[
[
T
]
]
R[[T]]
is not even cyclically pure in
S
[
[
T
]
]
S[[T]]
.