Abstract
Behn showed that if
K
[
G
]
is a prime group algebra with
G polycyclic-by-finite, then
K
[
G
]
is a
CS-ring if and only if
K
[
G
]
is a pp-ring if and only if
G is torsion-free or
G
≅
D
∞
and
char
(
K
)
≠
2
. As a consequence, such a group algebra
K
[
G
]
is hereditary excepting possibly when
K
[
G
]
is a domain. In this paper we show that if
K
[
G
]
is a semiprime group algebra of polycyclic-by-finite group
G and if
K
[
G
]
has no direct summands that are domains, then
K
[
G
]
is a
CS-ring if and only if
K
[
G
]
is hereditary if and only if
G
/
Δ
+
(
G
)
≅
D
∞
and
char
(
K
)
≠
2
. Precise structure of a semiprime
CS group algebra
K
[
G
]
of polycyclic-by-finite group
G, when
K is algebraically closed, is also provided.