In this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. Let
$A$
be a finitely generated commutative
$K$
–algebra over a field of characteristic 0, and let
$\sigma$
be a
$K$
–algebra automorphism of
$A$
. Given ideals
$I$
and
$J$
of
$A$
, we show that the set
$S$
of integers
$m$
such that
${{\sigma }^{m}}(I)\,\supseteq \,J$
is a finite union of complete doubly infinite arithmetic progressions in
$m$
, up to the addition of a finite set. Alternatively, this result states that for an affine scheme
$X$
of finite type over
$K$
, an automorphism
$\sigma \,\in \,\text{Au}{{\text{t}}_{k}}(X)$
, and
$Y$
and
$Z$
any two closed subschemes of
$X$
, the set of integers
$m$
with
${{\sigma }^{m}}(Z)\,\subseteq \,Y$
is as above. We present examples showing that this result may fail to hold if the affine scheme
$X$
is not of finite type, or if
$X$
is of finite type but the field
$K$
has positive characteristic.