1.π-projective modules. 2.Local modules. 3.Direct projective modules. 4.Refinable modules. 5.Properties of refinable modules. 6.Supplemented modules. 7.Properties of (f-) supplemented modules. 8.Finitely lifting modules. 9.Lifting modules. 10.π-projective lifting modules. 11.Decomposition of π-projective supplemented modules.
12.Projective semiperfect modules. 13.Projective f-semiperfect modules. 14.Perfect modules. 15.Uniserial modules.
We begin with weak projectivity conditions which are of interest for the investi-gations to follow.
Definitons. An A-module M is called π-projective if, for any two submodules U, V ⊂M with U +V =M, the epimorphism U ⊕V →M, (u, v)7→u+v, splits.
M is calleddirect projectiveif, for any direct summandXofM, every epimorphism M →X splits.
It is easy to see that M is π-projective if and only if for any two submodules U, V ⊂M with U +V =M, there exists f ∈End(M) with
Im(f)⊂U and Im(idM −f)⊂V. From [40, 41.14] we recall:
8.1 Properties of π-projective modules.
Assume M to be a π-projective A-module. Then:
(1) Every direct summand of M is π-projective.
(2) If M =U +V and U is a direct summand in M, then there exists V0 ⊂V with M =U ⊕V0.
(3) If M =U ⊕V, then V is U-projective (and U is V-projective).
(4) If M =U+V and U, V are direct summands in M, then U∩V is also a direct summand in M.
A moduleM is called localif it has a largest proper submodule. It is obvious that this submodule has to be equal to the radical of M and that Rad(M) M. Local modules are triviallyπ-projective.
8.2 Local modules. Characterizations.
For an non-zero A-module M, the following properties are equivalent:
(a) M is local;
(b) M has a maximal submodule which is superfluous;
(c) M is cyclic and every proper submodule is superfluous;
(d) M is cyclic and for any m, k ∈M, Am=M impliesAk =M orA(m−k) =M; (e) M is finitely generated and non-zero factor modules of M are indecomposable.
If M is projective in σ[M], then (a)-(c) are equivalent to:
(e) M is a projective cover of a simple module in σ[M];
(f ) End(AM) is a local ring.
Proof. Most of the assertions follow from [40, 19.7 and 41.4]. It only remains to show (c)⇔(d) Assume (c) and consider m, k ∈M with Am=M. Then
Ak+A(m−k) = M and we conclude Ak =M orA(m−k) = M.
Now assume (d) andU +V =M for submodules U, V ⊂M. Thenu+v =m for some u ∈ U, v ∈ V and so either u or m−u = v generate M. Hence every proper
submodule ofM is superfluous. 2
For direct projective modules we recall from [40, 41.18 and 41.19]:
8.3 Direct projective modules.
Assume M is a direct projective A-module and B =End(AM). Then:
(1) For direct summands U, V ⊂M, every epimorphism U →V splits.
(2) For direct summands U, V ⊂M with U +V =M, U ∩V is a direct summand in U (and M) and M =U ⊕V0 for some V0 ⊂V.
(3) K(B) :={f ∈B|Imf M} ⊂Jac(B).
(4) The following assertions are equivalent:
(a) For every f ∈B, there is a decomposition M =X⊕Y with X ⊂Imf and Y ∩Imf Y;
(b) B is f-semiperfect and K(B) =Jac(B).
(5) If M is a local module, then B is a local ring.
Definitions. AnA-moduleM is called refinable(or suitable) if for any submod-ulesU, V ⊂M withU+V =M, there exists a direct summand U0 ofM with U0 ⊂U and U0+V =M.
M is said to be strongly refinable if, in the given situation, there exist U0 ⊂ U, V0 ⊂V of M with M =U0⊕V0.
Notice that a finitely generated module M is (strongly) refinable if the defining conditions are satisfied for finitely generated submodules. So, for example, a finitely generated module M with every finitely generated submodule a direct summand is refinable (7.3).
We say thatdirect summands lift modulo a submoduleK ⊂M if under the canon-ical projection M → M/K every direct summand of M/K is an image of a direct summand of M. The connection between the above notions is evident from the next result.
8.4 Ref inable modules.
(1) For an A-module M, the following are equivalent:
(a) M is refinable;
(b) direct summands lift modulo every submodule of M. (2) The following assertions are also equivalent:
(a) M is strongly refinable;
(b) finite decompositions lift modulo every submodule ofM.
Proof. (1) (a)⇒(b) Let K ⊂M be a submodule. Assume M/K =U/K⊕V /K for submodules K ⊂U, V ⊂M. Then M =U +V and there is a direct summand U0 of M with U0 ⊂U and U0+V =M. This implies (U0 +K)/K =U/K.
(b)⇒(a) Now assumeM =U+V and putK =U∩V. ThenM/K =U/K⊕V /K and, by assumption, U/K lifts to a direct summand U0 of M. Since (U0 +K)/K = U/K we have U0+K =U, i.e., U0 ⊂U and M =U0+V +K =U0+V.
(2) The proof for strongly refinable modules is similar to the above. 2 8.5 Properties of ref inable modules.
Let M be an A-module and B =End(M).
(1) If M is (strongly) refinable and X ⊂ M is a fully invariant submodule, then M/X is (strongly) refinable.
(2) Every π-projective refinable module is strongly refinable.
(3) As an (A, B)-bimodule, M is refinable if and only if it is strongly refinable.
Proof. (1) Assume M/X = U/X +V /X for submodules U, V ⊂ M containing X. Then M = U +V and there exists a direct summand U0 of M with U0 ⊂ U and U0 +V = M. Assume M = U0 ⊕ W. Since X is fully invariant we have X = (X ∩ U0)⊕(X ∩W) and hence U0/(X ∩ U0) is a direct summand of M/X with the properties desired.
The same proof shows the assertion for strongly refinable modules.
(2) This follows from 8.1(2).
(3) AssumeM is a refinable (A, B)-bimodule and M =U+V with fully invariant submoduleU, V ⊂M. Then there exists a central idempotent e∈B, such that
M =M e+V and M e⊂U.
Multiplying the left equality withidM−e, we obtainM(idM−e) =V(idM−e)⊂V,
showing that M is strongly refinable. 2
Notice that finitely generated self-projective modules are refinable if and only if they have the exchange property (see Nicholson [212, Proposition 2.9]). Additional characterizations of refinable modules will be given in 18.7 and 18.10.
Definitions. LetU be a submodule of the A-module M. A submodule V ⊂M is called a supplement of U in M if V is minimal with the property U +V =M. It is easy to see that V is a supplement ofU if and only if
U +V =M and U ∩V V.
We sayU ⊂M has ample supplements in Mif for everyV ⊂M withU+V =M, there is a supplementV0 of U with V0 ⊂V.
M is called(amply) supplemented, if every submodule has (ample) supplements in M. Similarly (amply) f-supplemented modules are defined as modules whose finitely generated submodules have (ample) supplements.
Trivially, local modules are (amply) supplemented. Characterizations of these modules are given in [40, 41.6]:
8.6 Supplemented modules.
For a finitely generated A-module M, the following are equivalent:
(a) M is supplemented;
(b) every maximal submodule of M has a supplement in M;
(c) M is an irredundant (finite) sum of local submodules.
From [40, 41.2 and 41.3] we recall:
8.7 Properties of (f-) supplemented modules.
Let M be an A-module.
(1) Assume M is supplemented. Then any finitely M-generated module is supple-mented and M/Rad(M) is semisimple.
(2) Assume M is f-supplemented.
(i) IfL⊂M is finitely generated or superfluous, thenM/Lis also f-supplemented.
(ii) If Rad(M) M, then finitely generated submodules of M/Rad(M) are di-rect summands.
Definition. We call anA-module M (finitely) lifting, provided for every (finitely generated) submodule U ⊂ M, there exists a direct summand X of M with X ⊂ U and U/X M/X.
From [40, 41.11 and 41.13] we have characterizations for finitely generated modules of this type showing that they are in particular refinable:
8.8 Finitely lifting modules.
For a finitely generated A-module M, the following are equivalent:
(a) M is a finitely lifting module;
(b) for every cyclic (finitely generated) submodule U ⊂M, there is a decomposition M =X⊕Y, where X ⊂U and U ∩Y Y;
(c) for any cyclic submodule U ⊂ M, there exists an idempotent e ∈ End(AM), such that M e⊂U and U(1−e)M(1−e);
(d) M is amply f-supplemented and supplements are direct summands;
(e) M is strongly refinable and in M/Rad(M) every finitely generated submodule is a direct summand.
Proof. The first equivalences come from [40, 41.11 and 41.13].
(d)⇒(e) This implication is obvious by 8.7.
(e)⇒(b) For any finitely generated submodule U ⊂M, there is a decompositon M/Rad(M) = U +Rad(M)/Rad(M)⊕V /Rad(M),
for some submoduleRad(M)⊂V ⊂M. Since direct summands lift moduloRad(M), there exists a direct summand Y0 ⊂ M with Y0 +Rad(M) = V, M = U +Y0 and U ∩Y0 ⊂Rad(M), implying U ∩Y0 Y0.
M being strongly refinable, there is a decomposition M = X ⊕Y with X ⊂ U and Y ⊂Y0. From above we know U ∩Y ⊂U ∩Y0 Y0 implying U ∩Y Y. 2
Now we characterize lifting modules.
8.9 Lifting modules.
For an A-module M, the following are equivalent:
(a) M is a lifting module;
(b) for every submodule U ⊂M, there is a decomposition M =X⊕Y, where X ⊂U and U∩Y Y;
(c) for every submodule U ⊂ M, there exists an idempotent e ∈ End(AM), such thatM e⊂U and U(idM −e)M(idM −e);
(d) M is amply supplemented and every supplement is a direct summand;
(e) M is strongly refinable and M/Rad(M) is semisimple.
Proof. The first equivalences come from [40, 41.11 and 41.12].
(d)⇒(e) This implication is obvious by 8.7.
(e)⇒(b) Repeat the proof of the corresponding implication in 8.8. 2
π-projective supplemented modules are in fact lifting. From [40, 41.15] we have:
8.10 π-projective lifting modules.
Let M be an A-module and B =End(AM).
(1) The following are equivalent:
(a) M is lifting and π-projective;
(b) M is supplemented and π-projective;
(c) M is amply supplemented and the intersection of mutual supplements is zero;
(d) M is lifting and for direct summands U, V ⊂ M with M =U +V, U ∩V is a direct summand;
(e) for any submodules U, V ⊂M with U+V =M, there exists an idempotent e∈B, such that
M e⊂U, M(idM −e)⊂V and U(idM −e)M(idM −e).
(2) As an (A, B)-bimodule, M is lifting if and only if it is supplemented and π-projective.
Proof. It remains to show (2). Assume M is lifting and U, V ⊂ M are submodules with U +V =M. By 8.9, there is a central idempotent e∈B with
M e⊂U and U(idM −e)M(idM −e).
This implies M =M e+V and M(idM−e) =V(idM −e)⊂V. So M isπ-projective
and supplemented by (1). 2
For finitely generated modules of the above type the decomposition theorem [40, 41.17] yields:
8.11 Decomposition of π-projective supplemented modules.
Assume M is a finitely generated, π-projective lifting A-module. Then:
(1) M =LΛLλ with local modules Lλ, and
(2) for every direct summand N of M, there exists a subset Λ0 ⊂ Λ, such that M = (LΛ0Lλ)⊕N.
(3) If M =PΛNλ is an irredundant sum with indecomposable Nλ, thenM =LΛNλ. Remarks. The notion lifting modules is, for example, used in Oshiro [217]. In Z¨oschinger [288] the same modules are called strongly complemented (see [288, Satz 3.1]) and there are various other names used by different authors. Though we will not use this terminology we recall, for the convenience of the reader, some notation applied in the literature (e.g., Mohamed-M¨uller [27]).
Consider the following conditions on an A-moduleM:
(D1) for every submodule U ⊂M, there is a decompositionM =X⊕Y, where X ⊂U and Y ∩U Y.
(D2) If U ⊂M is a submodule such that M/U is isomorphic to a direct summand of M, thenU is a direct summand of M.
(D3) If U, V ⊂ M are direct summands with U +V = M, then U ∩V is a direct summand of M.
Condition (D1) characterizes lifting modules (see 8.9), (D2) is equivalent to direct projective. A moduleM with (D1) and (D2) is calleddual continuousordiscrete, and M is called quasi-discrete or quasi-semiperfect (in [216]) if it satisfies (D1) and (D3).
Quasi-discrete modules are just π-projective supplemented modules (e.g., [40, 41.15], [288, Satz 5.1]). The decomposition theorem for these modules mentioned above is originally due to Oshiro [216] and is also proved in [27, Theorem 4.15].
Dual continuous modules are direct projective lifting modules. Some of their properties are described in [27, Lemma 4.27] and [40, 41.18 and 41.19]. In particular they are quasi-discrete.
Dual to the notions considered above there are extending, π-injective and con-tinuous modules. For a detailed account on these we refer to [40], [27] and [11].
Definitions. N ∈σ[M] is said to besemiperfect in σ[M] if every factor module of N has a projective cover in σ[M].
N is f-semiperfect in σ[M] if, for every finitely generated submoduleK ⊂N, the factor module N/K has a projective cover in σ[M]. Obviously, finitely generated, self-projective local modulesM are (f-) semiperfect in σ[M].
Supplemented and semiperfect modules are closely related ([40, 42.5, 42.12]):
8.12 Projective semiperfect modules.
Assume the A-module M is projective in σ[M]. Then the following are equivalent:
(a) M is semiperfect in σ[M];
(b) M is supplemented;
(c) every finitely M-generated module has a projective cover in σ[M];
(d) (i) M/Rad(M) is semisimple and Rad(M)M, and (ii) decompositions of M/Rad(M) lift modulo Rad(M);
(e) M is a direct sum of local modules and Rad(M)M. If M is a finitely generated A-module, (a)-(f ) are equivalent to:
(g) End(AM) is a semiperfect ring.
For f-semiperfect modules we get from [40, 42.10]:
8.13 Projective f-semiperfect modules.
For a finitely generated, self-projective A-module M, the following are equivalent:
(a) M is f-semiperfect in σ[M];
(b) M is f-supplemented;
(c) (i) M/Rad(M) is regular in σ[M/Rad(M)], and (ii) decompositions lift modulo Rad(M).
M is said to beperfect in σ[M] if, for any index set Λ,M(Λ)is semiperfect inσ[M].
By [40, 43.8] we have a series of characterizations of these modules.
8.14 Perfect modules.
Let M be a finitely generated, self-projective A-module with endomorphism ring B =End(AM). The following statements are equivalent:
(a) M is perfect in σ[M];
(b) every (indecomposable) M-generated flat module is projective in σ[M];
(c) M(IN) is semiperfect in σ[M];
(d) M/Rad(M) is semisimple and Rad(M(IN))M(IN);
(e) B/Jac(B) is left semisimple and Jac(B) is right t-nilpotent;
(f ) B satisfies the descending chain condition for cyclic right ideals;
(g) BB is perfect in B-Mod.
For M =A, the above list characterizes left perfect algebras.
An R-module N is called uniserial if its submodules are linerarly ordered by in-clusion. N is serial if it is a direct sum of uniserial modules. Recall from [40, 55.1]:
8.15 Uniserial modules.
For an A-module N the following are equivalent:
(a) The (cyclic) submodules of N are linearly ordered;
(b) for every finitely generated non-zero submodule K ⊂N, K/Rad(K) is simple;
(c) for every factor module L of N, SocL is simple or zero.
References: Burkholder [102], Fuller [139], Mohamed-M¨uller [27], Nicholson [212], Oshiro [216, 217]. Wisbauer [40, 276], Z¨oschinger [288].