Let
I
I
be an
m
\mathfrak {m}
-primary ideal in a Gorenstein local ring (
A
A
,
m
\mathfrak {m}
) with
dim
A
=
d
\dim A = d
, and assume that
I
I
contains a parameter ideal
Q
Q
in
A
A
as a reduction. We say that
I
I
is a good ideal in
A
A
if
G
=
∑
n
≥
0
I
n
/
I
n
+
1
G = \sum _{n \geq 0} I^{n}/I^{n+1}
is a Gorenstein ring with
a
(
G
)
=
1
−
d
\mathrm {a} (G) = 1 - d
. The associated graded ring
G
G
of
I
I
is a Gorenstein ring with
a
(
G
)
=
−
d
\mathrm {a}(G) = -d
if and only if
I
=
Q
I = Q
. Hence good ideals in our sense are good ones next to the parameter ideals
Q
Q
in
A
A
. A basic theory of good ideals is developed in this paper. We have that
I
I
is a good ideal in
A
A
if and only if
I
2
=
Q
I
I^{2} = QI
and
I
=
Q
:
I
I = Q : I
. First a criterion for finite-dimensional Gorenstein graded algebras
A
A
over fields
k
k
to have nonempty sets
X
A
\mathcal {X}_{A}
of good ideals will be given. Second in the case where
d
=
1
d = 1
we will give a correspondence theorem between the set
X
A
\mathcal {X}_{A}
and the set
Y
A
\mathcal {Y}_{A}
of certain overrings of
A
A
. A characterization of good ideals in the case where
d
=
2
d = 2
will be given in terms of the goodness in their powers. Thanks to Kato’s Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set
X
A
\mathcal {X}_{A}
of good ideals in
A
A
heavily depends on
d
=
dim
A
d = \dim A
. The set
X
A
\mathcal {X}_{A}
may be empty if
d
≤
2
d \leq 2
, while
X
A
\mathcal {X}_{A}
is necessarily infinite if
d
≥
3
d \geq 3
and
A
A
contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring
k
[
X
1
,
X
2
,
X
3
]
k[X_{1},X_{2},X_{3}]
in three variables over a field
k
k
. Examples are given to illustrate the theorems.