1.Examples. 2.Non-M-singular modules. 3.Proposition. 4.Proposition. 5.Goldie tor-sion class. 6.Quotient ring ofA. 7.Corollary. 8.Rational submodules. 9.Maximal ring of quotients. 10.Exercises.
In torsion theory in A-Mod, an important part is played by the singular mod-ules. This notion can be adapted to the category σ[M] and many of the interesting properties are preserved.
Definition. Let M and N be A-modules. N is called singular in σ[M] or M-singularif N 'L/K for some L∈σ[M] and KL (see [273]).
In caseM =A, instead ofA-singularwe just saysingular. It is well-known that a module N is singular in A-Modif and only if for every n∈N, AnA(n) is an essential left ideal in A.
Obviously, every M-singular module is singular but not vice-versa as we will see with the examples below.
We denote by SM the class of singular modules in σ[M]. SM is closed under submodules, homomorphic images and direct sums (see [40, 17.3, 17.4]), i.e., it is a hereditary pretorsion class. Hence every module N ∈ σ[M] contains a largest M -singular submodule
SM(N) := Tr(SM, N).
However, SM need not be closed under extensions, i.e., SM(N/SM(N)) might not be zero. In our notationS(N) :=SA(N) is just the largest singular submodule ofN and SM(N)⊂ S(N).
IfTA is a hereditary pretorsion class in A-Mod, then the restriction ofTA toσ[M]
yields a (hereditary) pretorsion class TM := TA∩σ[M] in σ[M]. If M ∈ TA, then TM =σ[M]. Pretorsion classes inσ[M] may be induced by different classes inA-Mod.
In particular, the restriction of the singular torsion class S in A-Mod to σ[M] need not yieldSM.
10.1 Examples.
(1) Let M be a simple singular A-module. Then σ[M]⊂ S and SM = 0.
(2) An A-module M is semisimple if and only if SM = 0.
(3) Let A be an SI-ring (singulars are injective) and choose M ∈A-Mod such that S =σ[M]. ThenSM = 0.
(4) Put Q= Q/ZZ. Then in ZZ-Mod, σ[Q] = S and σ[Q] = SQ, i.e., every module inσ[Q] isQ-singular. This implies that σ[Q] has no projectives.
Proof. (1) All modules in σ[M] are singular in A-Mod and projective in σ[M].
(2) If M is semisimple, every module in σ[M] is projective and SM = 0.
Assume SM = 0. Then M has no non-trivial essential submodules and hence is semisimple.
(3) For an SI-ring, all singular modules inA-Mod are semisimple (see [11, 17.4]).
(4)σ[Q] is the category of torsionZZ-modules. To prove that every module inσ[Q]
isQ-singular it suffices to show this for all cyclic modulesZZ/pnZZ,n ∈ZZ, pa prime (a set of generators in σ[Q]). However, this is obvious by
ZZ/pnZZ '(ZZ/pn+1ZZ)/(pnZZ/pn+1ZZ) and pnZZ/pn+1ZZ ZZ/pn+1ZZ.
This provides a simple proof for the fact that there are no projective objects in σ[Q]
(see [40, 18.12]). 2
If SM(N) = 0, N is called non-singular in σ[M] or non-M-singular. Obviously, N is non-M-singular if and only if, for any K ∈ σ[M] and 0 6= f : K → N, Kef is not essential inK.
10.2 Non-M-singular modules.
Let M be a module with M-injective hull Mc.
(1) A module N in σ[M] with HomA(N,Mc) = 0 isM-singular.
(2) Assume SM(M) = 0. Then:
(i) N ∈σ[M] is M-singular if and only if HomA(N,Mc) = 0.
(ii) The class of M-singular modules is closed under extensions.
(iii) For all N ∈σ[M], SM(N/SM(N)) = 0.
Proof. (1) Consider N ∈ σ[M] which is not M-singular, i.e., SM(N) 6= N. Then the M-injective hull Nc is also not M-singular. Since it is M-generated, there is a morphism f :M →Ncwhose kernel is not essential in M.
The kernel of the restriction ¯f = f|N f−1 : N f−1 → N again is not an essential submodule and there exists a non-zero submodule K ⊂ N f−1 with K ∩Kef¯= 0.
Now ¯f|K :K →N is monic and the inclusionK ⊂Mc can be extended to a non-zero morphism inHomA(N,M).c
(2) (i)SM(M) = 0 impliesSM(Mc) = 0 and hence for everyM-singularN inσ[M], HomA(N,Mc) = 0. The other assertion follows from (1).
(ii) Let 0→K →L→N →0 be exact inσ[M] withK andN bothM-singular.
By (i), HomA(K,Mc) = HomA(N,Mc) = 0. Applying the functor HomA(−,Mc) to the exact sequence we see HomA(L,Mc) = 0 and L is M-singular by (i).
(iii) This is a consequence of (ii). 2
Next we collect some basic information on M-singular modules (see [11, 4.2,4.3]).
10.3 Proposition. Let M be an A-module.
(1) Any simple A-module is M-singular or M-projective.
(2) Every M-singular module is an essential submodule of an M-generated M -singular module.
(3) Each finitely generated M-singular module belongs toσ[M/L], for some LM. (4) {M/K|K M} is a generating set for the M-generated M-singular modules.
Proof. (1) LetEbe a simpleA-module. IfE 6∈σ[M] thenEis triviallyM-projective.
Assume E ∈σ[M] is not M-singular and consider an exact sequence 0−→K −→L−→E −→0
in σ[M]. By assumption, the maximal submodule K ⊂ L is not essential and hence is a direct summand inL, i.e., the sequence splits and E is projective in σ[M].
(2) Consider L∈ σ[M] and KL. The M-injective hull Lb of L is M-generated and
L/K ⊂L/K,b KLL.b
The inclusion map ofL/K into itsM-injective hull can be extended to L/Kb →L/K.d The image of this map is an essential M-generated M-singular extension of L/K.
(3) A finitely generated M-singular module is of the form N/K, for a finitely generated N ∈ σ[M] and K N. N is an essential submodule of a finitely M-generated moduleNf, i.e., there exists an epimorphism g :Mk →Nf,k ∈IN (compare (2)), and U := (N)g−1 and V := (K)g−1 are essential submodules of Mk.
With the canonical inclusions εi : M → Mk we get that L := Ti≤kV ε−1i is an essential submodule of M and Lk lies in the kernel of the composed map
U −→g N −→N/K.
This impliesN/K ∈σ[M/L].
(4) is an immediate consequence of (3). 2
10.4 Proposition. Let N be an M-singular module and f ∈HomA(M, N).
(1) If M is self-projective and (M)f is finitely generated, then Kef M. (2) If M is projective in σ[M], then KefM.
Proof. (1) Under the given conditions we may assume (M)f =L/K with L∈σ[M]
finitely generated and K L. Since M is self-projective it is also L-projective, and the diagram with the canonical projection p,
M
↓f
L −→p L/K −→ 0,
can be extended commutatively by some g :M →L. Then Kef = (K)g−1M.
(2) The arguments in (1) apply without finiteness condition. 2 As mentioned before, the class SM is not closed under extensions. We extendSM to a torsion class in the following way:
10.5 Goldie torsion class.
Let SM2 denote the modules X ∈ σ[M], for which there exists an exact sequence 0→K →X →L→0, with K, L∈ SM. Then:
(1) SM2 is a stable torsion class, called the Goldie torsion class in σ[M].
(2) For any N ∈σ[M], SM2 (N) = 0 if and only if SM(N) = 0, and
Proof. (1) It is straightforward to verify thatSM2 is closed under submodules, factors and direct summands. To show that it is closed under extensions, consider an exact sequence
Now it is obvious that SM2 is closed under essential extensions in σ[M].
(2) The first assertion is clear since every non-zero module in SM2 contains a non-zero submodule ofSM. Obviously, SM2 (N)/SM(N)⊃ SM(N/SM(N)). Consider M-injective and hence is a direct summand inQS2
M(N). SinceN/SM2 (N)⊂ImQS2
M(g) is an essential submodule in QS2
M(N), we concludeImQS2
M(g) =QS2
M(N).
This completes the proof that QS2
M(−) is exact. 2
The following assertion indicates a connection to the quotient ring of A. We consider the Goldie torsion theory inA-Mod and put S2 :=SA2.
10.6 Quotient ring of A.
Let M be an A-module with A∈σ[M]. Then:
(1) QS2(A) is a ring and QS2(M) is a generator in QS2(A)-Mod.
(2) QS2(M) is finitely generated and projective as module over T :=EndA(QS2(M))(=EndQ
S2(A)(QS2(M)).
(3) QS2(A)'End(QS2(M)T).
Proof. (1) The ring structure of QS2(A) is given in 9.20.
By assumption, A ⊂ Mk, k ∈ IN. Since QS2(−) is (left) exact, we conclude QS2(A)⊂QS2(M)k. By 9.13, QS2(A) is self-injective and hence is a direct summand inQS2(M)k, i.e., it is generated byQS2(M).
(2) and (3) characterize QS2(M) as generator in QS2(A)-Mod. 2 10.7 Corollary. Let M be a faithful module with SM(M) = 0. Then the following
are equivalent:
(a) A∈σ[M];
(b) Mc is finitely generated over EndA(Mc).
Proof. (a)⇒(b) Apply 10.6.
(b)⇒(a)AMc is a faithfulA-module and henceA ∈σ[Mc] =σ[M] (see [40, 15.4]).
2 The hereditary torsion theory in σ[M], whose torsionfree class is cogenerated by theM-injective hullMcofM, is called theLambek torsion theory inσ[M]. The torsion class is given by
TM ={K ∈σ[M]|HomA(K,Mc) = 0}.
Obviously, M is torsionfree in this torsion theory. In fact, TM is the largest torsion classe for which M is torsionfree. QTM(M) is just the (M,TM)-injective hull of M.
A TM-dense submodule U of N ∈σ[M] is said to be(M-) rational in N and N is called arational extension of U.
10.8 Rational submodules.
For a submodule U ⊂N with N ∈σ[M], the following are equivalent:
(a) U is rational in N;
(b) for any U ⊂V ⊂N, HomA(V /U, M) = 0.
Proof. (a)⇒(b) For 06=f ∈HomA(V /U, M), the diagram
0 → V /U → N/U
↓f
M → Mc
can be extended commutatively by a non-zero morphism N/U →Mc.
(b)⇒(a) Consider a non-zero g ∈ HomA(N/U,Mc) and M g−1 = V /U for some U ⊂V ⊂N. Then the restriction g|V /U ∈HomA(V /U, M) is not zero. 2 10.9 Maximal ring of quotients.
For M =A we have the torsion class in A-Mod,
T :=TA={K ∈A-Mod|Hom(K,A) = 0}.b
This yields the Lambek torsion theory inA-Mod. The quotient modulQT(A) allows a ring structure for whichAis a subring (see 9.20). QT(A) is called themaximal (left) quotient ring of A, denoted by Qmax(A).
For a non-M-singular M (i.e., a SM2 -torsionfree M), the Lambek torsion theory in σ[M] is closely related to the singular torsion theory. We will investigate these modules in the next section.
10.10 Exercises.
(1) Let M be an A-module. For any two classes of modules C and D in σ[M], denote by EM(D,C) the class ofA-modulesN for which there is an exact sequence
0→C →N →D→0 inσ[M], where C ∈C and D∈D.
(i) Let C and D be subclasses of σ[M]. Prove: If C and D are closed under sub-modules (factor sub-modules, direct sums) then EM(D,C) is also closed under submodules (resp. factor modules, direct sums).
(ii) Let C and D be subclasses of A-Mod which are closed under isomorphisms.
Prove: For any left ideal I ⊂A, A/I ∈ EA(D,C) if and only if there exists a left ideal J ⊃I of A such that J/I ∈C and A/J ∈D.
(2) Let M be a generator in A-Mod which is cogenerated by Ab (the injective hull in A-Mod). Put S = EndA(M) and T = EndA(Mc). Prove that the following are equivalent ([164, Theorem 3.5]):
(a) T is left self-injective and isomorphic toQmax(S);
(b) ST is isomorphic to the injective hull ofSS;
(c) HomA(M /M,c Mc) = 0.
(3) Let Qmax(A) be the maximal left quotient ring of A. We ask for properties of Qmax(A) as right A-module. Prove that the following are equivalent ([194]):
(a) Every finitely generated submodule of AQmax(A) is cogenerated by A;
(b) Qmax(A) is a rational extension of AA.
References. Berning [78], Dung-Huynh-Smith-Wisbauer [11], Izawa [164], Ma-saike [194], Wisbauer [273, 279].