Let
S
S
be a semigroup of matrices over a field such that a power of each element lies in a subgroup (i.e., each element has a Drazin inverse within the semigroup). The main theorem of this paper is that there exist ideals
I
0
,
…
,
I
t
{I_0}, \ldots ,{I_t}
of
S
S
such that
I
0
⊆
⋯
⊆
I
t
=
S
{I_0} \subseteq \cdots \subseteq {I_t} = S
,
I
0
{I_0}
is completely simple, and each Rees factor semigroup
I
k
/
I
k
−
1
{I_k}/{I_{k - 1}}
,
k
=
1
,
…
,
t
k = 1, \ldots ,t
, is either completely
0
0
-simple or else a nilpotent semigroup. The basic technique is to study the Zariski closure of
S
S
, which is a linear algebraic semigroup.