Let
A
=
A
(
k
)
A = A(k)
be the first Weyl algebra over an infinite field
k
k
, let
P
P
be any noncyclic, projective right ideal of
A
A
and set
S
=
End
(
P
)
S = \operatorname {End} (P)
. We prove that, as
k
k
-algebras,
S
≇
A
S\not \cong A
. In contrast, there exists a noncyclic, projective right ideal
Q
Q
of
S
S
such that
S
≅
End
(
Q
)
S \cong \operatorname {End} (Q)
. Thus, despite the fact that they are Morita equivalent,
S
S
and
A
A
have surprisingly different properties. For example, under the canonical maps,
Aut
k
(
A
)
≅
Pic
k
(
A
)
≅
Pic
k
(
S
)
{\operatorname {Aut} _k}(A) \cong {\operatorname {Pic} _k}(A) \cong {\operatorname {Pic} _k}(S)
. In contrast,
Aut
k
(
S
)
{\operatorname {Aut} _k}(S)
has infinite index in
Pic
k
(
S
)
{\operatorname {Pic} _k}(S)
.