τ-complemented and τ-supplemented modules
Proper classes of monomorphisms and short exact sequences were introduced by Buchsbaum to study relative homological algebra. It was observed in abelian group theory that complement submodules induce a proper class of monomorphisms and this observations were extended to modules by Stenstr¨om, Gene...
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irk-123456789-1573592019-06-21T01:28:10Z τ-complemented and τ-supplemented modules Al-Takhman, K. Lomp, C. Wisbauer, R. Proper classes of monomorphisms and short exact sequences were introduced by Buchsbaum to study relative homological algebra. It was observed in abelian group theory that complement submodules induce a proper class of monomorphisms and this observations were extended to modules by Stenstr¨om, Generalov, and others. In this note we consider complements and supplements with respect to (idempotent) radicals and study the related proper classes of short exact sequences. 2006 Article τ-complemented and τ-supplemented modules / K. Al-Takhman, C. Lomp, R. Wisbauer // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 1–15. — Бібліогр.: 17 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16D90, 16E50, 16.E99. http://dspace.nbuv.gov.ua/handle/123456789/157359 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Proper classes of monomorphisms and short exact
sequences were introduced by Buchsbaum to study relative homological algebra. It was observed in abelian group theory that complement submodules induce a proper class of monomorphisms and
this observations were extended to modules by Stenstr¨om, Generalov, and others. In this note we consider complements and supplements with respect to (idempotent) radicals and study the related
proper classes of short exact sequences. |
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Al-Takhman, K. Lomp, C. Wisbauer, R. |
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Al-Takhman, K. Lomp, C. Wisbauer, R. τ-complemented and τ-supplemented modules Algebra and Discrete Mathematics |
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Al-Takhman, K. Lomp, C. Wisbauer, R. |
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Al-Takhman, K. |
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τ-complemented and τ-supplemented modules |
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τ-complemented and τ-supplemented modules |
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τ-complemented and τ-supplemented modules |
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τ-complemented and τ-supplemented modules |
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τ-complemented and τ-supplemented modules |
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τ-complemented and τ-supplemented modules |
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Інститут прикладної математики і механіки НАН України |
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τ-complemented and τ-supplemented modules / K. Al-Takhman, C. Lomp, R. Wisbauer // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 1–15. — Бібліогр.: 17 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT altakhmank tcomplementedandtsupplementedmodules AT lompc tcomplementedandtsupplementedmodules AT wisbauerr tcomplementedandtsupplementedmodules |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2006). pp. 1 – 15
c© Journal “Algebra and Discrete Mathematics”
τ -complemented and τ -supplemented modules
Khaled Al-Takhman, Christian Lomp, Robert Wisbauer
Abstract. Proper classes of monomorphisms and short exact
sequences were introduced by Buchsbaum to study relative homo-
logical algebra. It was observed in abelian group theory that com-
plement submodules induce a proper class of monomorphisms and
this observations were extended to modules by Stenström, Gener-
alov, and others. In this note we consider complements and supple-
ments with respect to (idempotent) radicals and study the related
proper classes of short exact sequences.
1. Proper classes and τ-supplements
With the intention of formalising the theory of Ext functors depending
on a specific choice of monomorphisms, Buchsbaum introduced in [4]
certain conditions on a class of monomorphisms which are needed to study
relative homological algebra. This lead to he notion of proper classes of
monomorphisms and short exact sequences. Well-known examples of such
classes are the pure exact sequences which can be defined by choosing a
class P of (finitley presented) modules and considering those sequences on
which Hom(P,−) is exact for each P ∈ P. These techniques are outlined,
for example, in Mishina and Skornjakov [10], Sklyarenko [13] and [16].
It was observed in abelian group theory that complement submodules
induce a proper class of short exact sequences and this motivated the
investigation of such questions for modules over arbitrary rings. First
results in this direction were obtained, for example, by Stenström [14] and
Generalov [5, 7]. For a more comprehensive presentation of the results
and the sources we refer to E. Mermut’s PhD thesis [9].
2000 Mathematics Subject Classification: 16D90, 16E50, 16.E99.
Key words and phrases: proper classes; neat and coneat submodules; τ -
supplemented, τ -complemented and τ -semiperfect modules.
2 Complemented and supplemented
Our interest in this approach to the structure theory of modules is
based on an observation mentioned in Stenström [14], namely that over
any ring, supplement submodules induce a proper class of short exact
sequences. This may help to bring some order in the variations and
generalisations of supplemented and lifting modules coming up recently
by involving properties from torsion theory.
1.1. Proper classes. Let E be a class of short exact sequences in σ[M ].
If
0 // K
f
// L
g
// N // 0
belongs to E, then f is called an E-mono and g is said to be an E-epi.
The class E is called proper if it satisfies the conditions
P.1 E is closed under isomorphisms;
P.2 E contains all splitting short exact sequences in σ[M ];
P.3 the class of E-monos is closed under composition;
if f ′, f are monos and f ′ ◦ f is an E-mono, then f is an E-mono;
P.4 the class of E-epis is closed under composition;
if g, g′ are epis and g ◦ g′ is an E-epi, then g is an E-epi.
The class of all splitting short exact sequences in σ[M ] is an example
of a proper class.
1.2. Purities. Let P be a class of modules in σ[M ]. Denote by E
P the
class of all short exact sequences in σ[M ] on which Hom(P,−) is exact for
each P ∈ P. It is straightforward to prove that E
P is a proper class. This
type of class is called projectively generated and its elements are called
P-pure sequences.
The "classical" purity is obtained by taking for P all finitely presented
modules in σ[M ]; for M = R this is the Cohn purity (e.g., [16, § 33]).
1.3. Copurities. Let Q be a class of modules in σ[M ]. Denote by EQ
the class of all short exact sequences in σ[M ] on which Hom(−, Q) is
exact for each Q ∈ Q. Then EQ is a proper class. This type of class is
called injectively generated and its elements are called Q-copure sequences
(see [16, § 38]).
1.4. Relative injectivity and projectivity. Let E be a proper class
of short exact sequences in σ[M ].
A module P ∈ σ[M ] is called E-projective if Hom(P,−) is exact on
all short exact sequences in E.
Dually, a module Q ∈ σ[M ] is called E-injective if Hom(−, Q) is exact
on all short exact sequences in E.
K. Al-Takhman, Ch. Lomp, R. Wisbauer 3
It follows from standard arguments that the class of all E-projective
modules is closed under direct sums and direct summands, and the class
of all E-injective modules is closed under direct products and direct sum-
mands.
1.5. Class of complement submodules. Let Ec be the class of short
exact sequences 0 → K → L → N → 0 in σ[M ] such that K is a
complement (closed) submodule of L. Then:
(1) Ec is a proper class in σ[M ].
(2) Every (semi-) simple module (in σ[M ]) is Ec-projective.
Proof. (1) The conditions P.1 and P.2 are easily verified.
P.3. Let K ⊆ L and L ⊆ N be closed submodules. Choosing K̂ ⊆
L̂ ⊆ N̂ , we have
K = K̂ ∩ L = K̂ ∩ L̂ ∩ N = K̂ ∩ N,
proving that K is closed in N .
Consider K ⊆ L ⊆ N . If K is closed in N , then K = K̂∩N = K̂∩L,
that is, K is closed in L.
P.4. The composition of two epimorphisms g and h can be presented
by the commutative diagram with exact rows and columns,
0 // K
=
��
// U
i
��
// U/K //
��
0
0 // K // L
gh
��
g
// L/K //
h
��
0
L/U
=
// L/U,
where U/K = Ke h, for some K ⊆ U ⊆ L.
Assume g and h to have closed kernels. Suppose that U = Ke gh is
not closed in L and denote by U the essential closure of U in L. Then
K is closed in U and Ke h = U/K E U/K, contradicting the assumption
that h has a closed kernel.
On the other hand, assume U = Ke gh to be closed. Suppose that
Keh has a proper essential extension V in L/K. Then U = (Ke h)g−1 E
(V )g−1, contradiction our condition on Ke h. This shows that Keh is a
closed submodule of L/K.
(2) Let K ⊆ L be a closed submodule and S ⊆ L/K any simple
submodule. Then S ≃ N/K for some submodule K ⊆ N ⊆ L. By
4 Complemented and supplemented
assumption, K is a maximal submodule and is not essential in N . Hence
the map N → N/K ≃ S splits showing that Hom(S, L) → Hom(S, L/K)
is surjective and that S is Ec-projective.
Since direct sums of Ec-projectives are again Ec-projective, every
semisimple module is Ec-projective.
1.6. τ-complement submodules. Let τ be an idempotent preradical
for σ[M ] with associated classes Tτ and Fτ . Then for a submodule K ⊆ L
where L ∈ σ[M ], the following are equivalent:
(a) every N ∈ Tτ is projective with respect to the projection L → L/K;
(b) there exists a submodule U ⊆ L such that
K ∩ U = 0 and τ(L/K) = (U + K)/K ≃ U ;
(c) there exists a submodule U ⊆ L such that
K ∩ U = 0 and τ(L/K) ⊆ (U + K)/K ≃ U.
If this conditions are satified, then K is called a τ -complement in L.
Proof. (a)⇒(b) A pullback construction yields the commutative diagram
with exact rows
0 // K
iK
//
=
��
K̃ //
��
τ(L/K) //
i
��
0
0 // K // L
g
// L/K // 0.
Since τ(L/K) ∈ Tτ , there exists a morphism h : τ(L/K) → L with
i = hg. By the Homotopy Lemma (e.g., [16, 7.16]), this implies that the
top row splits, that is,
K̃ = K ⊕ U, for some U ⊆ K̃ and (K + U)/K = τ(L/K).
(b)⇒(a) Let N ∈ Tτ and f ∈ Hom(N, L/K). Then Im f ⊆ τ(L/K)
and it can be seen from the diagram in the proof of (a)⇒(b) that there
is a morphism h : N → L with f = hg.
(b)⇔(c) This is easy to verify.
1.7. Corollary. Let τ be a preradical for σ[M ] and K ⊆ L where L ∈
σ[M ].
(1) If K is a τ -complement in L and L/K ∈ Tτ , then K is a direct
summand.
K. Al-Takhman, Ch. Lomp, R. Wisbauer 5
(2) If L ∈ Tτ , then every τ -complement submodule of L is a direct
summand.
For the preradical induced by the class of all (semi-)simple modules
we find an interesting relationship with the complement submodules.
1.8. Neat submodules. A monomorphism f : K → L is called neat if
any simple module S is projective relative to L → L/Im f , that is, the
Hom sequence Hom(S, L) → Hom(S, L/K) → 0 is exact. The class of
short exact sequences with neat monomorphisms is a projectively gener-
ated class in the sense of 1.2.
As shown in 1.5, all sequences in Ec are neat.
1.9. When neat submodules are closed in σ[M ]. For a module M
the following are equivalent:
(a) every neat submodule of M is closed;
(b) a submodule of M is closed if and only if it is neat;
(c) for every L ∈ σ[M ], closed submodules of L are neat;
(d) for every essential submodule U ⊆ M , Soc M/U 6= 0;
(e) every M -singular module is semi-artinian.
Proof. (a)⇔(b) is clear since closed submodules are neat.
(c)⇒(a) is obvious.
(a)⇒(d) Let U E M be a proper submodule. Then U is not closed
and hence not neat in M . Thus there exists a morphism g : S → M/U
where S is simple, that can not be extended to a morphism S → M . In
particular, this implies that Im g 6= 0, that is, Soc M/U 6= 0.
(d)⇒(e) Let U ⊆ V ⊆ M . If U EM then V EM and hence (d) implies
that every factor module of M/U has nonzero socle, that is, M/U is semi-
artinian (see [6, 3.12]).
By [6, Proposition 4.3], the set {M/U |U E M} is a generating set
of all M -singular M -generated modules and every M -singular module is
a submodule of M -generated M -singular modules. Thus if all the M/U
have nonzero socles then this is also true for all M -singular modules.
(e)⇒(c) Let K ⊆ L be a neat submodule and assume that it has a
proper essential extension K ⊆ L. Then K/K is an M -singular module
and hence, by assumption, contains a simple submodule S = N/K where
K EN ⊆ K. Now neatness of K ⊆ L implies that the map N → N/K =
S splits. This contradicts K E N proving that K is closed in L.
For M = R we obtain the following characterisation which was (partly)
proved in [5, Theorem 5]:
6 Complemented and supplemented
1.10. When neat submodules are closed in R-Mod. For a ring R
the following are equivalent:
(a) every neat left ideal of R is closed;
(b) a left ideal of R is closed if and only if it is neat;
(c) for every left R-module, closed submodules are neat;
(d) for every essential left ideal I ⊆ R, Soc R/I 6= 0;
(e) every singular module is semi-artinian.
Rings with these properties are called C-rings (in [12]).
Dualising the notions considered above yields the following.
1.11. τ-supplement submodules. Let τ be a radical for σ[M ] with
associated classes Tτ and Fτ . Then for a submodule K ⊆ L where L ∈
σ[M ], the following are equivalent:
(a) every N ∈ Fτ is injective with respect to the inclusion K → L;
(b) there exists a submodule U ⊆ L such that
K + U = L and U ∩ K = τ(K);
(c) there exists a submodule U ⊆ L such that
K + U = L and U ∩ K ⊆ τ(K).
If this conditions are satisfied, then K is called a τ -supplement in L.
Proof. (a)⇒(b) Consider the commutative diagram with exact rows
0 // K
i
//
p
��
L //
��
L/K //
=
��
0
0 // K/τ(K) // L/τ(K) // L/K // 0.
Since K/τ(K) ∈ Fτ , there exists h : L → K/τ(K) with p = ih. By
the Homotopy Lemma (e.g., [16, 7.16]), this implies that the bottom row
splits, that is,
L/τ(K) = K/τ(K) ⊕ U/τ(K), for some τ(K) ⊆ U ⊆ L.
This means L = K + U and U ∩ K = τ(K).
(b)⇒(a) By the given data, L/τ(K) = K/τ(K) ⊕ U/τ(K). Let
N ∈ Fτ and f ∈ Hom(K, N). Then τ(K) ⊆ Ke f and we have the
K. Al-Takhman, Ch. Lomp, R. Wisbauer 7
commutative diagram
0 // K
i
//
p
��
f
��
L //
��
L/K //
=
��
0
K/τ(K) //
f̄
zzvv
v
v
v
v
v
v
v
L/τ(K) // L/K // 0.
N
Since the middle row splits we obtain a morphism h : L → N with f = ih.
(b)⇔(c) One direction is trivial.
Assume K + U = L and K ∩ U ⊆ τ(K). Putting U ′ = U + τ(K) we
have
K + U ′ = L and K ∩ U ′ = K ∩ U + τ(K) = τ(K).
1.12. Corollary. Let τ be a radical for σ[M ] and K ⊆ L where L ∈
σ[M ].
(1) If K is a τ -supplement in L and K ∈ Fτ , then K is a direct sum-
mand.
(2) If L ∈ Fτ , then every τ -supplement submodule of L is a direct
summand.
(3) If K is a τ -supplement in L and X ⊆ K, then K/X is a τ -
supplement in L/X.
Proof. (1) and (2) follow from the preceding observations. (3) Let U ⊆ L
be such that K +U = L and K ∩U ⊆ τ(K). Then K/X +(U +X)/X =
L/X and
K ∩ (U + X)/X = (K ∩ U + X)/X ⊆ (τ(K) + X)/X ⊆ τ(K/X).
As a special case we consider the radical (for σ[M ]) cogenerated by
the simple modules.
1.13. Co-neat submodules. A monomorphism f : K → L is called
co-neat if any module Q with Rad Q = 0 is injective relative to it, that
is, the Hom sequence Hom(L, Q) → Hom(K, Q) → 0 is exact. The class
of short exact sequences with co-neat monomorphisms is an injectively
generated class in the sense of 1.3.
8 Complemented and supplemented
1.14. Characterisation of co-neat submodules. For a submodule
K ⊆ L, the following are equivalent:
(a) K → L is a co-neat submodule;
(b) there exists a submodule U ⊆ L such that
K + U = L and U ∩ K = Rad K;
(c) there exists a submodule U ⊆ L such that
K + U = L and U ∩ K ⊆ Rad K.
If these conditions are satisfied, then K is a Rad-supplement in L. If
RadK ≪ K, then K is co-neat (Rad-supplement) in L if and only if it is
a supplement in L (see [9]).
1.15 Lemma. A small submodule N of a module L is co-neat in L if
and only if RadN = N .
Proof. Let N ≪ L. If N is co-neat in L, then there exists K ⊆ L such
that N + K = L and N ∩ K = Rad N . Since N ≪ L, K = L and
hence N = N ∩ L = RadN . On the other hand assume N ≪ L and
Rad N = N . Then N + L = L and N ∩ L = N = RadN , thus N is a
co-neat submodule of L.
1.16. When are co-neat submodules coclosed. Let M be a module.
Then the following conditions are equivalent:
(a) every non-zero co-neat submodule of a module in σ[M ] is a coclosed
submodule;
(b) every non-zero M -small module in σ[M ] is a Max module (resp.
has a maximal submodule).
Proof. (a)⇒(b) If N ≪ L and Rad N = N , then, by the Lemma 1.15, N
is a co-neat submodule of L. But by hypothesis co-neat submodules are
coclosed submodules and hence not small - a contradiction.
(b)⇒(a) Let N be a co-neat submodule of a module L ∈ σ[M ]. Then
for any submodule U ⊆ N , N/U is co-neat in M/U . To see this let K
be a submodule of L such that N + K = L and N ∩K = Rad(N). Then
N/U + (K + U)/U = L/U and
N/U (K + U)/U = ((N K) + U)/U = (RadN + U)/U ⊆ Rad N/U.
Hence N/U is co-neat in L/U . Suppose N/U ≪ L/U , then Rad N/U =
N/U by the Lemma 1.15. But by hypothesis N/U is a Max module,
and hence has a proper maximal submodule. Hence N/U 6≪ L/U for all
U ⊂ N implies that N is coclosed in L.
K. Al-Takhman, Ch. Lomp, R. Wisbauer 9
Let M be a cosemisimple module. Then Rad N = 0 for any N ∈
σ[M ]. Hence any non-zero submodule of a module in σ[M ] is coclosed.
Moreover any non-zero module is a Max module. However a submodule
N of a module L is co-neat if and only if it is a direct summand. Thus
if M is not semisimple, there are coclosed submodules which are not co-
neat. This shows that in general the dual statement of the statement for
neat submodules (1.9 (b)⇔(e)) does not hold.
2. τ-supplemented modules
Throughout this section τ will denote a radical for σ[M ]. Recall that a
submodule K ⊆ L is called a τ -supplement provided there exists some
U ⊆ L such that U + K = L and U ∩ K ⊆ τ(K) (1.11). To some
extent the theory of supplemented, lifting and semiperfect modules can
be transferred to the corresponding notions based on τ -supplements. This
will be sketched in this section.
2.1. Definition. A module L is said to be τ -supplemented if every
submodule K ⊆ L has a τ -supplement in L, and it is called amply τ -
supplemented if for any submodules K, V ⊆ L such that K + V = L,
there is a τ -supplement U for K with U ⊆ V .
2.2. τ-supplemented modules. Let L be a τ -supplemented module in
σ[M ].
(1) Every submodule K ⊆ L with K ∩ τ(L) = 0 is a direct summand.
In particular, if L is τ -torsion-free, then L is semisimple.
(2) Every factor module and every direct summand of L is τ -supplemented.
(3) L/τ(L) is a semisimple module.
(4) L = U ⊕ N where N is semisimple and τ(U) E U .
Proof. (1) Recall that τ(K) ⊆ K ∩ τ(L) and then refer to 1.12.
(2) and (3) are also obvious consequences of 1.12.
(4) Let N ⊆ L be a complement for τ(L), i.e. N ∩ τ(L) = 0 and
N ⊕ τ(L) E L. This implies τ(N) = 0. By assumption, there exists
U ⊆ L such that N + U = L and N ∩ U ⊆ τ(U). By construction,
N ∩ U = N ∩ (N ∩ U) ⊆ N ∩ τ(U) ⊆ N ∩ τ(L) = 0,
hence L = N⊕U and τ(L) = τ(N)⊕τ(U) = τ(U). Thus N⊕τ(U)EN⊕U
and this implies τ(U) E U . By (1), N is semisimple.
2.3. Sums of τ-supplemented modules. Let L ∈ σ[M ].
10 Complemented and supplemented
(1) Let L1, U ⊆ L be submodules where L1 is τ -supplemented. If L1+U
has a τ -supplement in L, then so does U .
(2) If L1 and L2 are τ -supplemented modules in σ[M ] and L = L1+L2,
then L is τ -supplemented.
(3) Any finite sum of τ -supplemented modules is τ -supplemented.
(4) If L is τ -supplemented, then every finitely L-generated module is
τ -supplemented.
Proof. (1) By assumption, there exists X ⊆ L such that (L1+U)+X = L
and (L1 + U)∩X ⊆ τ(X). Now (U + X)∩L1 has a τ -supplement in L1,
that is, some Y ⊆ L1 with
(U + X) ∩ L1 + Y = L1 and (U + X) ∩ Y ⊆ τ(Y ).
Since U + X + Y = L we have that Y is a τ -supplement of U + X
in L. To prove that X + Y is a τ -supplement of U in L is remains to
show that U ∩ (X + Y ) ⊆ τ(X + Y ). Now Y + U ⊆ L1 + U , hence
X ∩ (Y + U) ⊆ X ∩ (L1 + U) ⊆ τ(X) and therefore
(X + Y )∩U ⊆ X ∩ (Y + U) + Y ∩ (X + U) ⊆ τ(X) + τ(Y ) ⊆ τ(X + Y ).
(2) Let U ⊆ L be any submodule. Then L1 + L2 + U = L trivially
has a τ -supplement in L and hence, by (1), L2 + U has a τ -supplement
in L. Again by (1), this implies that U has a τ -supplement in L. Thus
L is a τ -supplemented module.
(3) and (4) are immediate consequences of (1) and 2.2(2).
2.4. Amply τ-supplemented modules. If L ∈ σ[M ] is an amply
τ -supplemented module, then
(1) direct summands of L are amply τ -supplemented and
(2) factor modules of L are amply τ -supplemented.
Proof. (1) Assume L = K ′ ⊕ K and let X, Y be submodules of K with
K = X + Y . Since L = K ′ + X + Y , there exists Y ′ ⊆ Y such that
Y ′+K ′+X = L and Y ′∩(K ′+X) ⊆ τ(Y ′). Now Y ′∩X ⊆ Y ′∩(X+K ′) ⊆
τ(Y ′) and
K = K ∩ L = K ∩ (Y ′ + (K ′ ⊕ X))
= K ∩ (K ′ ⊕ X) + Y ′ = X + Y ′.
Hence Y ′ is a τ -supplement of X in K and Y ′ ⊆ Y .
(2) Assume X ⊆ L and L/X = K/X + K ′/X, where X ⊆ K ⊆ L,
X ⊆ K ′ ⊆ L. Since L = K+K ′, there exists Y ⊆ K ′ such that K+Y = L
and K ∩ Y ⊆ τ(Y ). We will show that Y + X/X is a τ -supplement of
K. Al-Takhman, Ch. Lomp, R. Wisbauer 11
K/X in L/X. It is clear that (Y + X)/X + K/X = L/X and that
(Y + X)/X ⊆ K ′/X. Also we have
K/X ∩ (Y + X)/X = ((K ∩ Y ) + X)/X
⊆ (τ(Y ) + X)/X ⊆ τ(Y + X/X).
2.5. Corollary. Let L be amply τ -supplemented in σ[M ].
(1) If K is a τ -supplement in L and K ∈ Fτ , then K is amply τ -
supplemented.
(2) If L ∈ Fτ , then every τ -supplement submodule of L is is amply
τ -supplemented.
Proof. (1) and (2) follow directly from 1.12.
An R-module L is called π-projective if for any submodules U, V of
L such that U + V = L, there exists f ∈ End(L) with Im(f) ⊆ U and
Im(1 − f) ⊆ V (see [16, 41.14] for details).
2.6. Proposition. Let L ∈ σ[M ].
(1) If every submodule of L is τ -supplemented, then L is an amply τ -
supplemented module.
(2) If L is π-projective and τ -supplemented, then L is amply τ -supple-
mented.
Proof. (1) Assume U, V ⊆ L such that L = U + V . Since U is τ -
supplemented, there exists Y ⊆ U such that U ∩ V + Y = U and also
U ∩ V ∩ Y ⊆ τ(Y ). We know that V ∩ Y = U ∩ V ∩ Y ⊆ τ(Y ) and it is
clear that L = Y + V .
(2) Assume L = X +Y , so there exists e ∈ End(L) such that Le ⊆ X,
L(1− e) ⊆ Y . It is clear that X(1− e) ⊆ X. Since L is τ -supplemented,
there exists C ⊆ L such that C + X = L and C ∩ X ⊆ τ(C). Hence
L = Le + L(1 − e)
= Le + (X + C)(1 − e) ⊆ X + C(1 − e).
Therefore L = X + C(1 − e). It can be shown easily that C(1 − e) ⊆ Y ,
and that X ∩C(1− e) = (X ∩C)(1− e). Since X ∩C ⊆ τ(C), it follows
that X ∩ C(1 − e) ⊆ τ(C(1 − e)). Hence C(1 − e) is a τ -supplement of
X in L and C(1 − e) ⊆ Y , i.e., L is amply τ -supplemented.
2.7. Corollary. For M the following are equivalent:
12 Complemented and supplemented
(a) every module in σ[M ] is τ -supplemented;
(b) every module in σ[M ] is amply τ -supplemented.
2.8. τ-dense summands. Let L ∈ σ[M ]. For a submodule U ⊆ L the
following are equivalent:
(a) there is a decomposition L = X ⊕ X ′ with X ⊆ U and X ′ ∩ U ⊆
τ(X ′);
(b) there is an idempotent e ∈ End(L) with Le ⊆ U and U(1 − e) ⊆
τ(U(1 − e));
(c) there is a direct summand X of L with X ⊆ U and U/X ⊆ τ(L/X);
(d) U has a τ -supplement V in L such that U ∩V is a direct summand
of U ;
(e) there is a decomposition U = X ⊕ Y , such that X is a direct sum-
mand of L and Y ⊆ τ(L).
In this case we say that U contains a τ -dense direct summand.
Proof. Compare [16, 41.11] and [15, Theorem 2.8].
2.9. τ-lifting modules. A module L ∈ σ[M ] is called τ -lifting if every
submodule of L contains a τ -dense direct summand.
This concept has been introduced by P. F. Smith and I. Al-Khazzi in
[1] for more general classes X of modules instead of a torsion theory τ .
2.10. Properties of τ-lifting modules. Let L ∈ σ[M ] be a τ -lifting
module. Then:
(1) Rad L ⊆ τ(L) and if RadL 6= τ(L), then L has a nonzero direct
summand that is τ -torsion.
(2) Any direct summand of L is τ -lifting.
Proof. (1) By 2.2(3), L/τ(L) is semisimple and hence RadL ⊆ τ(L).
Suppose Rad L 6= τ(L). Then there is a maximal submodule K ⊆ L
with τ(L) 6⊆ K. By assumption, K contains a τ -dense summand, that
is a submodule A ⊆ K with L = A ⊕ B and K ∩ B ⊆ τ(B) and K ∩ B
is a maximal submodule in B. Thus τ(B) = B or K ∩ B = τ(B). In
the latter case τ(L) = τ(A) ⊕ τ(B) ⊆ K, contradicting the choice of K.
Thus L = A ⊕ B where B = τ(B).
(2) Assume L = K ⊕ K ′ and let X ⊆ K. Since L is τ -lifting, L =
N ⊕ N ′ with N ⊆ X and X ∩ N ′ ⊆ τ(N ′). So K = N ⊕ (K ∩ N ′) and
X ∩ (K ∩ N ′) = X ∩ N ′ ⊆ τ(N ′)
⊆ τ(L) = τ(K) ⊕ τ(K ′) = τ(N) ⊕ τ(K ∩ N ′) ⊕ τ(K ′).
But X ∩N ′ ⊆ K ∩N ′, hence X ∩N ′ ⊆ τ(K ∩N ′). So K is τ -lifting.
K. Al-Takhman, Ch. Lomp, R. Wisbauer 13
2.11. τ-covers. An epimorphism in σ[M ] f : P → L is called a τ -cover
provided Ke f ⊆ τ(P ). If P is projective in σ[M ], then f is called a
projective τ -cover (in σ[M ]).
The notion of (projective) Rad-covers was studied in [3, 17] under the
name generalised (projective) covers. Nakahara [11] also studied ρ-covers
under the name ρ-semicovers, for a (normal) preradical ρ.
The following lemma is needed to show general properties of τ -covers
(compare [2, 8.17]).
2.12. Lemma. If f : L → N is an epimorphism such that Ker(f) ⊆
τ(L), then f(τ(L)) = τ(N).
2.13. Properties of (projective) τ-covers.
(1) If f : P → L is a projective τ -cover and g : L → N is a τ -cover,
then gf : P → N is a projective τ -cover.
(2) If each fi : Pi → Li, i ∈ I is a (projective) τ -cover, then the map
⊕Ifi :
⊕
I Pi →
⊕
I Li is a (projective) τ -cover.
Proof. (1) To show that Ke gf ⊆ τ(P ), let x ∈ Ke gf , so f(x) ∈ Ke g ⊆
τ(L) = f(τ(P )), hence f(x) = f(t1), where t1 ∈ τ(P ), which implies
that x − t1 ∈ Ke f ⊆ τ(P ), therefore x ∈ τ(P ).
(2) This follows directly from the facts that τ commutes with direct
sums and that a direct sum of projective modules is projective.
From the definition, the following is clear: If K ⊆ L is a τ -supplement
for U ⊆ L, then K → K/K ∩ U ∼= L/U is a τ -cover.
Now we give a connection between τ -covers and τ -supplements.
2.14. Projective τ-cover and τ-supplements. For U ⊆ L with L in
σ[M ] the following are equivalent:
(a) L/U has a projective τ -cover;
(b) U has a τ -supplement V which has a projective τ -cover;
(c) if V ⊆ L and L = U + V , then U has a τ -supplement V ′ ⊆ V such
that V ′ has a projective τ -cover.
Proof. (a)⇒(b) Assume that f : P → L/U is a projective τ -cover of
L/U . Then there exists g : P → L such that f = πg where π : L →
L/U denotes the canonical projection. Denote by V the image of g, i.e.
V = Im(g). Since g−1(U) = Ker f , we have Ker g ⊆ Ker f , i.e. P is a
projective τ -cover of V . As U ∩ V = g(Ker f) ⊆ g(τ(P )) ⊆ τ(g(P )) =
τ(V ), we have V is a τ -supplement of U in L having the τ -projective
cover P .
14 Complemented and supplemented
(b)⇒(a) Let V be a τ -supplement of U in L having a projective τ -
cover f : P → V . Consider g : V → V/U ∩ V = L/U , this is a τ -cover
of L/U . By 2.13 it follows that gf : P → L/U is a projective τ -cover of
L/U .
(a)⇒(c) Assume that f : P → L/U is a τ -cover and P is projective.
Whenever L = U + V , then V/(V ∩U) ≃ L/U and P is also a projective
τ -cover of V/(V ∩ U). By (2), there exists V ′ ⊆ V such that V ′ is a
τ -supplement of V ∩U in V , having a projective τ -cover. Since V ′ ∩U =
V ′ ∩ (V ∩ U) and
L = V + U = V ′ + (V ∩ U) + U = V ′ + U,
V ′ is a τ -supplement of U in L.
(c)⇒(b) follows from L = U + L.
2.15. τ-semiperfect. A module L ∈ σ[M ] is called τ -semiperfect (τ -
perfect), if every factor module of L (any direct sum of copies of L) has
a projective τ -cover.
From the above results we obtain the
2.16. Characterisation of τ-semiperfect modules. For a module
L ∈ σ[M ] the following are equivalent:
(a) L is τ -semiperfect;
(b) L is τ -supplemented by supplements which have projective τ -covers;
(c) L is amply τ -supplemented by supplements which have projective
τ -covers.
2.17. Corollary. Let L ∈ σ[M ] be τ -semiperfect, then
(1) L/τ(L) is semisimple;
(2) if L is also τ -torsion-free, then L is semisimple
If M = R and L is projective in R-Mod then it is Rad-semiperfect
(Rad-perfect) if and only if it is semiperfect (perfect). This was estab-
lished in [3, 11, 17].
Acknowledgment. This paper was initiated during a visit of K.
Al-Takhman at the Heinrich Heine University of Düsseldorf. He wishes
to thank the Department of Mathematics for the hospitality and also the
DAAD, German Academic Exchange Service, for the financial support.
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Contact information
Khaled
Al-Takhman
Department of Mathematics, Birzeit Uni-
versity, Birzeit P.O. Box 14, Palestine
E-Mail: takhman@birzeit.edu
Christian Lomp Centro de Matematica da Universidade do
Porto, 4169-007 Porto, Portugal
E-Mail: clomp@fc.up.pt
16 Complemented and supplemented
Robert Wisbauer Mathematisches Institut der, Heinrich-
Heine Universität, 0225 Düsseldorf,
Germany
E-Mail: wisbauer@math.uni-duesseldorf.de
Received by the editors: 16.01.2006
and in final form 21.11.2006.
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