The Divisible and E-Injective Hulls of a Torsion Free Group
Recently there has been considerable interest in the structure of torsion free abelian groups G as modules over their endomorphism rings E = End(G). In particular, homological properties of the E-module G have been the focus of a number of papers (see (2)
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Recently
there
has
been considerable
structure of torsion free abelian their
endomorphism
homological
06268
rings
properties of
E
=
groups
interest
In G
finite rank
groups
G
particular,
have been
focus of a number of papers (see (2), (7), (8)). structure of
the
G as modules over
End( G).
the E-module
in
such that
the
In ( 2) the G
was
E-
projective was determined, while in (7) the injective hull of G as an E-module was investigated. question was asked: hull of
G equal
For which groups to
QG,
equivalent to the question: module?
In the latter paper the
the
G is the E-injective
divisible hull?
When is
QG
This is
injective as a QE-
In this note we obtain a partial answer by imposing
an additional
condition on
which
injective as a QE-module
QG
is
Q(center E).
G.
We examine groups and
G for
HomQE(QG,QG)
=
This property, called aEqi, is equivalent .to a
R. Göbel et al. (eds.), Abelian Groups and Modules © Springer-Verlag Wien 1984
164
C. Vinsonhaler
condition weaker than quasi-injectivity on The
strongly indecomposable
EG.
finite rank
aEqi groups
G
turn out to be exactly those with a commutative, local, selfinjective quasi-endomorphism ring for which rank G (Proposition 1.3). that any
finite
It
rank E(G)
=
is also shown (Corollary 1.4)
dimensional ,
local,
commutative
injective rational algebra arises in this way.
self-
The arbitrary
finite rank aEqi groups are classified in Theorem 2.8, giving a large class of groups
G such
that
QG
is injective as a
QE-module. 0. Definitions and preliminaries.
Throughout, group means
the letter
torsion free
ring of the group
G
the center of
If
represents the
E.
G always denotes a group, abelian group. E(G)
is written X
group or
X is torsion free.
quasi-endomorphism ring of
The G,
is regarded as a left module over We employ the standard
E,
or
Q ®2 X,
regarded as a subgroup or subring since
The endomorphism
is a group or ring, ring
of ring and E,
where
and QX.
C
is
the symbol
QX
X
is
always
This is possible
QE(G) G,
and
is called the
respectively
respectively
QG,
QE.
definitions of quasi-isomorphism,
quasi-equality and strongly indecomposable.
The Divisible and £-Injective Hulls
165
Our initial proposition,
stated for reference,
is due to
J.D. Reid. Let
Proposition 0.1.
G be
a finite rank
group.
The
following are equivalent: 1)
G
is strongly indecomposable,
2)
QE(G)
3)
Any endomorphism of
is local,
G to be almost
Define a group provided that homomorphism
given any
there is
g:G
such that
~G
Note that the map
g
E-quasi-injective (aEqi)
E-submodule
f:H -> G,
E-homomorphism
the center of
G is monic or nilpotent.
of
H
G,
and
E-
n
and
a positive integer g
lifts
nf.
may be regarded as an element of
C,
The next lemma restates the definition of
E.
aEqi in more familiar terms. Lemma 0.2. (a)
QG
(b)
HomQE(QG,QG)
K'
Let
K be a QE-submodule of
a QE-homomorphism.
K such that for each
=~ i
is aEqi if and only if
= QC.
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