1.Projective modules. 2.Self-generators. 3.M as a generator inσ[M]. 4.Density Theo-rem. 5.Generators inA-Mod. 6.Examples. 7.Intrinsically projective modules. 8.Prop-erties of intrinsically projective modules. 9.Ideal modules. 10.Self-progenerator.
11.Properties of the trace. 12.The socle of a module. 13.Exercises.
5.1 Projective modules.
Let M and P be A-modules. P is said to be M-projective if every diagram in A-Mod with exact row
P
↓
M −→g N −→ 0
can be extended commutatively by some morphism P →M.
So, P isM-projective if and only if HomA(P,−) is exact with respect to all exact sequences 0→K →M →N →0 in A-Mod.
If M is M-projective, then it is also calledself- (or quasi-) projective.
Assume M is self-projective and K ⊂ M is a fully invariant submodule, i.e., for any f ∈End(AM), Kf ⊂K. ThenM/K is also self-projective.
A finitely generated self-projective module M is also N-projective for every N ∈ σ[M], i.e., it is projective in σ[M] (see [40, Section 18]).
A submodule K of M is superfluousor smallinM, written K M, if, for every submoduleL⊂M, K+L=M impliesL=M.
Let N ∈ σ[M]. By a projective cover of N in σ[M] we mean an epimorphism π:P →N with P projective in σ[M] andKe π P.
Definitions. M is called a self-generator if it generates all its submodules. A module N ∈σ[M] is a generator in σ[M] if N generates all modules in σ[M].
5.2 Self-generators.
For an A-module M with B =End(AM), the following are equivalent:
(a) M is a self-generator;
(b) for every A-submodule U ⊂M, U =M HomA(M, U);
(c) the mapU 7→HomA(M, U)from submodules of M to left ideals ofB is injective.
Assume M is a self-generator. Then:
(1) For any m∈M, β ∈End(MB), there exists a∈A with β(m) =am.
Hence A-submodules of M are precisely its End(MB)-submodules.
(2) If K ⊂M is a fully invariant submodule, then M/K is also a self-generator.
Proof. The equivalences are easily derived from the fact that a submodule U ⊂M isM-generated if and only if U =M HomA(M, U).
(1) follows from the proof of the Density Theorem (e.g., [40, 15.7]).
(2) A submodule inM/Kis of the formL/K, for some submoduleLwithK ⊂L⊂ M. By assumption, there exists an epimorphism M(Λ) →L. Since every component of this map is an endomorphism of M, we see that the submodule K(Λ) ⊂ M(Λ) is contained in the kernel of the epimorphism
M(Λ) →L→L/K.
Hence L/K is generated by M/K. 2
The following properties of generators are shown in [40, 15.5 and 15.9].
5.3 M as a generator in σ[M].
For an A-module M with B =End(AM), the following are equivalent:
(a) M is a generator in σ[M];
(b) the functor HomA(M,−) :σ[M]→B-Mod is faithful;
(c) M generates every (cyclic) submodule of M(IN); (d) M(IN) is a self-generator;
(e) for every (cyclic) submodule U ⊂M(IN), U =MHomA(M, U).
If M is a generator in σ[M], then M is a flat End(AM)-module.
There is another important property of generators and cogenerators (see 6.3) in σ[M] shown in [40, 15.7 and 15.8]:
5.4 Density Theorem.
Let M be an A-module with B =End(AM)having one of the following properties:
(i) M is a generator in σ[M], or
(ii) for any cyclic submodule U ⊂ Mn, n ∈ IN, the factor module Mn/U is cogenerated by M. Then:
(1) For any finitely many m1, . . . , mn in M and α ∈ End(MB), there exists a ∈ A with α(mi) = ami for all i= 1, . . . , n.
In this caseA/AnA(M) is said to be adense subring of End(MB).
(2) Every A-submodule of M(Λ) is an End(MB)-submodule of M(Λ).
(3) If M is finitely generated over End(AM), then A/AnA(M)'End(MB).
Generators in the full module category have additional chacterizations:
5.5 Generators in A-Mod.
Let M be an A-module and B =End(AM).
(1) M is a generator in A-Mod if and only if (i) MB is finitely generated and B-projective;
(ii) A'End(MB).
(2) Assume M is a generator in A-Mod andB is commutative. Then M is a finitely generated and projective A-module.
Proof. (1) See, for example, [40, 18.8].
(2) By (1), MB is finitely generated, projective and faithful. SinceB is commuta-tive this implies that MB is a generator in Mod-B (see [40, 18.11]). Now applying (1) to the moduleMB, we conclude thatM is finitely generated and projective as module
overA'End(MB). 2
Studying the relationship between submodules of M and left ideals of the endo-morphism ring a weak form of projectivity (introduced in Brodskii [90]) turns out to be of interest:
An A-module M is said to be intrinsically projective if every diagram with exact row
M
↓
Mn −→ N −→ 0,
wheren ∈IN and N ⊂M, can be extended commutatively by some M →Mn. M is calledsemi-projectiveif the above condition (only) holds forn = 1. As easily seen, M is semi-projective if and only if for every cyclic left ideal I ⊂ End(AM), I =HomA(M, M I) (see [40], before 31.10).
Of course every self-projective module is intrinsically projective. However there are also other types of examples:
5.6 Examples. Let M be an A-module with B =End(AM).
(1) If kernels of endomorphisms of M are M-generated andB is a left PP ring, then M is semi-projective.
(2) If MB is flat and B is left semihereditary, thenM is intrinsically projective.
(3) If B is a regular ring, then M is intrinsically projective.
Proof. (1) For any f ∈ B, Bf is projective, and by [40, 39.10], T r(M,Kef) is a direct summand in M. By our conditionT r(M,Kef) = Kef.
(2) Since MB is flat, the kernel of any g :Mk→Mk,k ∈IN, is M-generated (see [40, 15.9]). Since B is left semihereditary, End(AMk) is left PP (semihereditary) by [40, 39.13] and we see from (1) that Keg is a direct summand in Mk.
From this we conclude that the exact sequence in the diagram for the definition of intrinsically projective is in fact splitting. Hence M is intrinsically projective.
(3) This is a special case of (2). 2
Now we turn to general properties of the modules just introduced.
5.7 Intrinsically projective modules.
For an A-module M with B =End(AM), the following are equivalent:
(a) M is intrinsically projective;
(b) I =HomA(M, M I) for every finitely generated left ideal I ⊂B.
If AM is finitely generated, then (a),(b) are equivalent to:
(c) I =HomA(M, M I) for every left ideal I ⊂B;
(d) the map I 7→M I from left ideals in B to submodules of M is injective.
Proof. (a)⇒(b) Consider a left ideal I ⊂B generated byγ1, . . . , γk ∈B. From the exact sequence
Mk
Pγ
−→i M I −→0 the functor HomA(M,−) yields the sequence
HomA(M, Mk)
PHom(M,γi)
−→ HomA(M, M I)−→0,
which is also exact by (a). Since Bk ' HomA(M, Mk), this means that γ1, . . . , γk generate HomA(M, M I) as B-module. HenceI =HomA(M, M I).
(b)⇒(a) Consider an exact sequence of A-modules
Mk−→f U −→0 with k ∈IN , U ⊂M.
With the canonical injections εi :M →Mk, we form the left ideal I =PBεif ⊂B.
Then U =M I and the lower row in the following diagram is exact HomA(M, Mk) Hom(M,f)−→ HomA(M, U)
↓' ↓'
Bk
Pεif
−→ I −→ 0.
From this we see thatHomA(M, f) is epic, i.e., M is intrinsically projective.
(a) ⇒(c) Assume M is intrinsically projective and finitely generated. Then it is easy to see thatHomA(M,−) is exact on all exact sequences
M(Λ) −→U −→0, where U ⊂M, Λ any set.
Hence the assertion is shown with the same proof as (a)⇒(b).
(c)⇒(a) and (c)⇒(d) are obvious.
(d)⇒(c) This is an immediate consequence of the equalityM I =MHomA(M, M I),
for any left idealI ⊂B. 2
5.8 Properties of intrinsically projective modules.
Let M be an intrinsically projective A-module, which is finitely generated by ele-ments m1, . . . , mk ∈M, and denote B =End(AM). Then the map
B →Mk, s 7→(m1, . . . , mk)s, is a monomorphism of right B-modules.
(1) For every left ideal I ⊂B, (m1, . . . , mk)B ∩MkI = (m1, . . . , mk)I. (2) For every proper left ideal I ⊂B, M I 6=M.
(3) If MB is flat, then (m1, . . . , mk)B is a pure submodule of MBk.
(4) If MB is projective, then B is isomorphic to a direct summand ofMBk and hence MB is a generator in Mod-B.
Proof. (1) Certainly the right hand side is contained in the left hand side.
Assume (m1, . . . , mk)f ∈ MkI = (M I)k for f ∈ B. Then M f =PiAmif ⊂M I and hence f ∈Hom(M, M I) = I (since M is intrinsically projective).
(2) This is an obvious consequence of (1)
(3) If MB is flat, then MBk is also flat and by (1), (m1, . . . , mk)B ' B is a pure submodule ofMBk (e.g., [40, 36.6]).
(4) SinceMBk is projective, the factor moduleMBk/(m1, . . . , mk)Bis pure-projective (see [40, 34.1]) and hence projective by (3). SoB is isomorphic to a direct summand
of MBk. 2
Definitions. LetA, B denote associative unital R-algebras. An (A, B)-bimodule M is said to be a B-ideal module if the map I 7→M I defines a bijection between the left ideals ofB and the A-submodules of M (with inverse K 7→ {b∈B |M b⊂K}).
An A-module M is called ideal module if - considered as (A,End(AM))module -M is anEnd(AM)-ideal module, i.e., if the mapI 7→M Iis a bijection between the left ideals of End(AM) and the A-submodules of M (with inverse K 7→HomA(M, K)).
For these modules we have interesting characterizations (compare [276, Proposition 2.6], [139, Lemma 2.4]):
5.9 Ideal modules.
Let A, B denote associative unitalR-algebras andM an(A, B)-bimodule, which is finitely generated as A-module.
(1) The following are equivalent:
(a) M is a B-ideal module;
(b) every submodule of AM is of the form M I for some left ideal I ⊂B, and MB is faithfully flat.
For B =End(AM), (a)-(b) are equivalent to:
(c) M is intrinsically projective and a self-generator.
(2) If M is a B-ideal module then for any multiplicative subset S ⊂R, the (AS−1, BS−1)-bimodule M S−1 is a BS−1-ideal module (see 16.5).
(3) Assume B =End(AM) and M is an ideal module. Then:
(i) B is isomorphic to a pure submodule of MBk, k∈IN. (ii) If B is right perfect, thenMB is a generator in Mod-B and
AM is a projective generator in σ[M].
Proof. (1) (a)⇒(b) We show that MB is faithfully flat. Without restriction, for any f ∈ B we may assume f ∈ End(AM) and we have the exact commutative diagram with canonical map µf
M ⊗BHomA(M, Ke f) → M ⊗BB → M ⊗BBf → 0
↓ ↓' ↓µf
0 → Ke f → M → M f → 0 .
Since Ke f is M-generated the first vertical map is epic and hence µf is an isomor-phism.
Using the relation M(I ∩J) = M I∩M J for left ideals I, J of B, it is straight-forward to show by induction (on the number of generators of I as B-module) that µI :M⊗BI →M I is an isomorphism for every finitely generated left idealI ⊂B and thereforeMB is flat. SinceM I 6=M for every proper left idealI ⊂B, the B-module MB is faithfully flat.
(b) ⇒ (a) We prove that I 7→ M I is injective. Assume M I =M J for left ideals I, J ⊂B. Without restriction we may assumeJ ⊂I. ThenM⊗BI/J 'M I/M J = 0 and hence I =J since MB is faithfully flat.
(a)⇔(c) follows immediately from 5.2 and 5.7.
(2) Assume M is aB-ideal module. Every AS−1-submodule U ⊂M S−1 is of the form U0S−1 for some submodule U0 ⊂ M (see 16.5). Since U0 = M I for some left ideal I ⊂B, we conclude
U =U0S−1 =M IS−1 =M S−1IS−1.
Since MB is a faithfully flat B-module, M S−1 is a faithfully flat BS−1-module.
Now the assertion follows from (1).
(3) (i) By (1), MB is a flat module and the assertion follows from 5.8(3).
(ii) Since B is right perfect, MB is projective (by (1)), and MB is a generator in Mod-B by 5.8(4). Hence M is (finitely generated and) projective over its biendo-morphism ringEnd(MB) (see 5.5). Since theA-module structure of M is identical to its End(MB)-module structure (see 5.2) we conclude that M is self-projective as an A-module and, by [40, 18.5], M is a projective generator in σ[M]. 2 Definitions. A finitely generated, projective generator N in σ[M] is called a progenerator in σ[M]. If M is a progenerator in σ[M] we call it a self-progenerator.
Of course, every self-progenerator is an ideal module.
The importance of this notion is clear by the following characterizations:
5.10 Self-progenerator.
For a finitely generated A-module M with B =End(AM), the following are equiv-alent:
(a) M is a self-progenerator;
(b) M is a generator in σ[M] and MB is faithfully flat;
(c) (i) for every left ideal I ⊂B, M I 6=M and
(ii) for every finitely M-generated A-module U, the canonical map M⊗BHomA(M, U)→U is injective (bijective);
(d) there are functorial isomorphisms (i) idB−M od'HomA(M, M⊗B−) and (ii) M ⊗BHomA(M,−)'idσ[M];
(e) HomA(M,−) :σ[M]→B-Mod is an equivalence of categories.
Proof. (a)⇔(b)⇔(e) follow from [40, 18.5 and 46.2].
For any A-submodule K ⊂ Mn, we put U = Mn/K and form the commutative diagram with exact lower row,
M ⊗BHomA(M, K) → M ⊗BHomA(M, Mn) → M ⊗BHomA(M, U) →0
↓ϕ1 ↓' ↓ϕ2
0→ K → Mn → U →0.
(a)⇒(c) SinceM is self-projective, the upper row is also exact, and by the Kernel Cokernel Lemma,ϕ2 is injective if and only if ϕ1 is surjective.
MB being flat, property (i) is equivalent toMB being faithfully flat. We know this from (a)⇔(b).
(c)⇒(b) Sinceϕ2is an isomorphism, the upper row is exact. Again by the Kernel Cokernel Lemma,ϕ1 is surjective andM is a generator in σ[M].
(d)⇔(e) This characterizes HomB(M,−) to be an equivalence (e.g., [40, Section
46]). 2
Let U be a class of A-modules and N anA-module. N is said to be U-generated, if there exists an epimorphism LΛUλ →N with Uλ ∈ U.
The sum of allU-generated submodules of N is called the trace of U in N, Tr(U, N) = X{Im h|h∈Hom(U, N), U ∈ U }.
IfU ={U} we simply writeTr(U, L) =Tr({U}, L).
From [40, 13.5] we obtain:
5.11 Properties of the trace.
Let U and N be A-modules.
(1) Tr(U, N) is the largest submodule of N generated by U. (2) N =Tr(U, N) if and only if N is U-generated.
(3) Tr(U, N) is an End(AN)-submodule of N.
As a special case we consider the trace of all simple modules in M. 5.12 The socle of a module.
The socle of an A-module M is defined as the sum of all simple (minimal) sub-modules of M and we have the characterization (e.g., [40, 21.1 and 21.2])
Soc(M) = P{K ⊂M |Kis a simple submodule inM}
= T{L⊂M|Lis an essential submodule inM}.
and the following properties:
(1) For any morphism f :M →N, Soc(M)f ⊂Soc(N).
(2) For any submodule K ⊂M, Soc(K) =K ∩Soc(M).
(3) Soc(M)M if and only if Soc(K)6= 0 for every non-zero submodule K ⊂M. (4) Soc(M) is also an End(AM)-submodule of M (fully invariant).
(5) Soc(LΛMλ) =LΛSoc(Mλ).
5.13 Exercises.
(1) LetM be a finitely generatedA-module which is a generator inσ[M]. Assume thatEnd(AM)is a left semihereditary ring. Prove thatM is self-projective (and hence a progenerator inσ[M]).
(2) Let M, N be A-modules and B =End(AM).
N is called restricted M-projective if the functor HomA(N,−) is exact on exact se-quences of the form
Mn→L→0, whereL⊂Mk,n, k ∈IN. Prove:
(i) If MB is FP-injective and N is restricted M-projective, then HomA(N, M)B is FP-injecticve.
(ii) If HomA(N, M)B is FP-injecticve, then N is restricted M-projective.
(iii) The following are equivalent:
(a) Every A-module is restricted M-projective;
(b) MB is flat and B is left semihereditary.
References: Brodskii [90], Fuller [139], Wisbauer [40, 275].