1.Polyform modules. 2.Properties of EndA(Mc). 3.Monoform modules. 4.Maximal quotients of non-singular rings. 5.Quotient rings of EndA(M). 6.Finite dimensional polyform modules. 7.Goldie’s Theorem. 8.Polyform subgenerators. 9.Trace and torsion submodules. 10.Properties. 11.Properties of polyform modules. 12.Bimod-ule properties of polyform mod12.Bimod-ules. 13.Independence over the endomorphism ring.
14.Self-injective polyform modules. 15.Idempotent closure of polyform modules. 16.Ex-ercises.
A moduleM is called polyformif every essential submodule is rational in M. Our next result shows that this is just another name for a non-M-singular module M. 11.1 Polyform modules. Characterization.
For anA-module M with M-injective hull M, the following are equivalent:c (a) Every essential submodule is rational in M;
(b) for any submodule K ⊂M and 06=f :K →M, Kef is not essential in K; (c) for any K ∈σ[M] and 06=f :K →M, Kef is not essential in K;
(d) M is non-M-singular (i.e., SM(M) = 0);
(e) EndA(Mc) is regular.
Proof. (a)⇒(e) Apply 9.16.
(a)⇒(d) Assume 0 6= K ⊂ M is an M-singular submodule. By 10.3, K is contained in an M-generated M-singular submodule Kf ⊂ Mc and, moreover, Kf is generated by{M/U |UM}. However, UM implies HomA(M/U,Mc) = 0. Hence K ⊂Kf= 0.
(c)⇔(d)⇒(b) are obvious.
(e)⇒(b) Suppose for some K ⊂M and f :K →M that KefK. Thenf can be extended to an ¯f ∈EndA(Mc) with Kef¯Mc. EndA(Mc) being regular, Kef¯is a direct summand (see 7.6) and hence ¯f and f have to be zero.
(b)⇒(a) Consider U M and f ∈ HomA(V /U, M), for any U ⊂ V ⊂ M. With the projection p : V → V /U we obtain pf ∈ HomA(V, M) which has an essential kernel. From (b) we conclude thatpf and f are zero. Now apply 10.8. 2 11.2 Properties of EndA(Md).
Let M be a polyform A-module and T =EndA(Mc).
(1) T is regular and left self-injectice.
(2) EndA(M) is subring of T.
(3) Every monomorphism f ∈EndA(M) with Imf M is invertible in T. (4) M has finite uniform dimension if and only if T is left semisimple.
Proof. Combine 11.1, 9.18 and 9.16. 2
A module M is said to be monoform if every submodule is rational inM. Notice that any uniform submodule of a polyform module is monoform. As a special case of 11.1 we obtain:
11.3 Monoform modules. Characterization.
For anA-module M with M-injective hull M, the following are equivalent:c (a) Every non-zero submodule is rational in M (M is monoform);
(b) for any submodule K ⊂M and 06=f :K →M, Kef = 0;
(c) M is uniform and non-M-singular (i.e., SM(M) = 0);
(d) EndA(Mc) is a division ring.
Let E(A) denote the A-injective hull of AA. Applying the preceding results to M =A we have:
11.4 Maximal quotients of non-singular rings.
Let A be a left non-singular ring.
(1) Qmax(A) =EndA(E(A)) is a selfinjective, regular ring.
(2) AA has finite uniform dimension if and only if Qmax(A) is left semisimple.
(3) Assume A is a subring in any left self-injective ring Q and A AQ. Then Q'Qmax(A).
(4) If AA is uniform then Qmax(A) is a division ring.
Proof. (1),(2) and (4) follow from 11.2 and 11.1.
(3) Since AAQ, we may assume AQ⊂Qmax(A). Denote by ·the multiplication inQand by∗the multiplication inQmax(A). We show thatp·q=p∗qfor allp, q ∈Q.
Consider K ={s∈A|sp ∈A}=Ap−1A. For every k∈K, k(p·q−p∗q) = (kp)q−(kp)q = 0,
i.e., K(p·q −p∗q) = 0. Since Qmax(A) is a non-singular A-module, this implies p·q =p∗q. So Qmax(A) is aQ-module and QQmax(A) is a direct summand, i.e.,
Q=Qmax(A). 2
An overring Q⊃ A is called a classical left quotient ring of A if every element in A, which is not a zero divisor, is invertible in Q and all elements of Q have the form s−1t, where s, t ∈Aand s is not a zero divisor. For details about these rings we refer to Lambek [183] or Stenstr¨om [39].
For a polyform moduleM, we know from 11.4 thatS =EndA(M) is subring of the self-injective, regular ring T =EndA(Mc). In general, T need not to be the maximal quotient ring ofS andS need not even be non-singular. This is the case under special conditions.
11.5 Quotient rings of EndA(M).
Let M be a polyform A-module with S =EndA(M).
(1) Assume for all non-zero submodules N ⊂M, HomA(M, N)6= 0. Then:
(i) T :=EndA(Mc) is an essential extension of SS.
(ii) S is left non-singular and T =Qmax(S).
(2) Assume for every KM there is a monomorphism g :M →K with M gK.
Then EndA(Mc) is a classical left quotient ring of S.
Proof. (1)(i) For a non-zerof ∈EndA(Mc),K =M f−1∩M is an essential submodule of M. FromHomA(M, N)6= 0 for all non-zero N ⊂M, we deduce
Tr(M, K) =MHomA(M, K)KM.
This implies MHomA(M, K)f 6= 0 and we can find some g ∈ HomA(M, K) with 06=gf ∈S. Thus gf ∈S∩Sf 6= 0.
(ii) Assume SS has a singular submodule Sb 6= 0, b ∈ S. Then K = AnS(b)
SSST. This implies that T K ⊂ AnT(b) are essential left ideals in T, i.e., T b is a singular submodule in TT. However, T is regular and so b ∈ S(TT) = 0, i.e., SS is non-singular. Therefore Qmax(S) is equal to theS-injective hull Sb of S.
Since TT is self-injective we can now apply 11.4(3).
(3) We show that any q∈EndA(Mc) has the formq =s−1t, for s, t∈S: Consider N :=M q−1∩M M and choose a monomorphism s:M →N with M sM. Now (the extension of) s is invertible inEndA(Mc) (see 11.2) and q=s−1t. 2 Under the general assumptions in 11.5, condition (2) holds for modules with finite uniform dimension and semiprime endomorphism ring. We cite this from [11, 5.19]:
11.6 Finite dimensional polyform modules.
Let M be an A-module with S = EndA(M). Assume HomA(M, N) 6= 0 for all non-zero N ⊂M. Then the following assertions are equivalent:
(a) M is polyform with finite uniform dimension and S is semiprime;
(b) M is polyform with finite uniform dimension and, for every N M, there is a monomorphism M →N;
(c) EndA(Mc) is left semisimple and is the classical left quotient ring of S.
In this case, for every essential submodule L of M, LEndA(Mc) = Mc.
In case S = EndA(M) is commutative, one of the conditions in 11.6(a) is auto-matically satisfied: From 11.5 we know that S is non-singular and any non-singular commutative ring is semiprime.
Applied to M =A, 11.6 yields the classical (see [11, 5.20])
11.7 Goldie’s Theorem. For a ring A, the following are equivalent:
(a) A is semiprime and AA is non-singular with finite uniform dimension;
(b) A is semiprime and AA has finite uniform dimension and acc on annihilators;
(c) A has a classical left quotient ring which is left semisimple.
11.8 Polyform subgenerators.
Let M be a polyform A-module, T =EndA(Mc) and suppose A ∈σ[M]. Then:
(1) A is left non-singular and QS(A) = Qmax(A).
(2) Mc is a generator in QS(A)-Mod.
(3) McT is finitely generated and T-projective, and QS(A)'EndT(Mc).
(4) If M has finite uniform dimension, then T and Qmax(A) are left semisimple.
Proof. (1),(4)A∈σ[M] impliesA ⊂Mk, for somek ∈IN. HenceAAis non-singular.
IfM has finite uniform dimension then AA has finite uniform dimension.
(2),(3) Apply 10.6. 2
For convenience we fix the following notation.
11.9 Trace and torsion submodules.
Let M be an A-module, Mc its self-injective hull and T =EndA(Mc). Let K ⊂Mc be a submodule and L∈σ[M].
By TK(L) we denote thetrace of σ[K] in L, i.e.
TK(L) =X{U ⊂L|U ∈σ[K]}.
For cyclic submodules Aa⊂L, put Ta(L) =TAa(L).
By TK we denote the hereditary torsion class in σ[M] determined by KT(⊂d Mc).
Then forL∈σ[M],TK(L) denotes the corresponding torsion submodule of L, i.e.
TK(L) =X{U ⊂L|HomA(U,KTd) = 0}.
In particular, for cyclic modules K =Aa, we put Ta(L) =TAa(L).
We list some properties related to these notions.
11.10 Properties. Let M be an A-module, K ⊂M and T =EndA(Mc).
(1) TK(Mc) =KT is a K-injective module.
(2) TK(M) =TK(Mc)∩M.
(3) for any submodule L⊂Mc, TK(L)∩ TK(L) = 0.
(4) For submodules L, N ⊂M with K =L+N,
TK(Mc) =TL(Mc) +TN(M)c and TK(M) = TL(M)∩ TN(M).
(5) If K is generated by a1, . . . , an∈K, TK(M) =Tni=1Tai(M).
Proof. (1) KT is injective in σ[K] (see [40, 16.8]).
(2) TK is a hereditary torsion class.
(3) For X ⊂ TK(L)∩ TK(L), X ⊂ KT and HomA(X,KTd) = 0. This implies X = 0.
(4) ForK =L+N,KT =LT+N T. Hence without restriction assume K =KT, L = LT and N = N T. Clearly TK(M) ⊂ TL(M)∩ TN(M). Consider an additive complement No ⊂N of L in K. ThenL⊕No is an essential submodule of K and
Kc=Lb⊕Nco ⊂Lb ⊕N .c From this we concludeTK(M)⊃ TL(M)∩ TN(M).
(5) This is derived from (4) by induction. 2
11.11 Properties of polyform modules.
Let M be a polyform A-module, Mc its M-injective hull, and T =EndA(Mc). Then for any submodule K ⊂M:
(1) TK(Mc) is M-injective.
(2) TT
K(M)b (Mc) =KTd. (3) Mc=TK(Mc)⊕KTd. (4) TK(M) +TK(M)M.
(5) for any m∈ Mc, AnT(m) is generated by an idempotent in T and Mc is a non-singular right T-module.
Proof. PutL=TK(Mc).
(1) Since Mc is polyform, HomA(L,b KTd) = 0 and hence Lb ⊂ L, i.e., Lb = L is M-injective.
(2) Consider g ∈ HomA(K, L). Since Keg is not essential in K, there exists 0 6= X ⊂ K with X ∩Keg = 0 and X ' (X)g ⊂ L, a contradiction. Hence K ⊂ TL(Mc) and KT ⊂ TL(Mc) since TL(Mc) is fully invariant.
Now we show that KT TL(Mc). Assume there is a non-zero submodule U ⊂ TL(Mc) withU ∩KT = 0. If HomA(U,KTd) = 0, then U ⊂L and
U ⊂ TL(Mc)∩ TL(M) = 0, a contradiction.c
Hence there is a non-zero g : U → KTd. Since Keg is not essential in U, there exists a non-zero submoduleV ⊂U withV ∩Keg = 0. Because ofKTKTd, we may assumeV '(V)g ⊂KT. Now for some t ∈T, we have V = (V)gt ⊂U∩KT = 0, a contradiction.
(3) By (1), L and TL(Mc) are M-injective. Since L∩ TL(M) = 0 by definiton,c Mc =L⊕ TL(Mc)⊕W for someW ⊂Mc.
Assume there is a non-zero h∈HomA(W, L) andQ∩Keh= 0, for some non-zero Q⊂W. ThenQ'(Q)h⊂L and there is somet ∈T with Q= (Q)ht⊂W ∩L= 0.
This impliesHomA(W, L) = 0 and W ⊂ TL(Mc).
Hence W = 0 and by (2), Mc=L⊕ TL(M) =c L⊕KTd. (4) is an immediate consequence of (3).
(5) For m ∈ Mc consider t ∈ T with (m)t = 0. Then (Am)t = 0 and - Mc being polyform - (Am)td = 0. We have Mc = Amd ⊕U for some A-submodule U ⊂Mc. For the related projection (idempotent) g : Mc → Am,d M gtc = 0. This means gt = 0 and t = (1−g)t ∈ (1−g)T. Therefore AnT(m) = (1−g)T implying that Mc is a
non-singularT-module. 2
As already shown above, the condition on anA-module to be polyform has a strong influence on the structure of its fully invariant submodules. We collect information about this in our next lemma:
11.12 Bimodule properties of polyform modules.
LetM be a polyformA-module, Mc its M-injective hull andT =EndA(Mc). Denote byC the centre ofT (i.e., the endomorphism ring ofMcas an(A, T)-bimodule). Then:
(1) Every essential (A, T)-submodule of Mc is essential as an A-submodule.
(2) Mc is self-injective and polyform as an (A, T)-bimodule.
C is a regular self-injective ring.
(3) For every submodule (subset) K ⊂ M, there exists an idempotentc ε(K) ∈ C, such that AnC(K) = (1−ε(K))C.
(4) If KL⊂Mc, then ε(K) =ε(L).
(5) Every finitely generated C-submodule of Mc is C-injective.
(6) If Mc is a finitely generated (A, T)-module, Mc is a generator in C-Mod.
Proof. (1) Let N ⊂ Mc be an essential (A, T)-submodule. Then N ∩ TN(M) = 0c impliesTN(Mc) = 0 and Mc=N Td =Ncby 11.11.
SoN Mc as an A-submodule.
(2) Again letN ⊂Mc be an essential (A, T)-submodule andh:N →Mc an (A, T )-morphism. Since Mc is a self-injective A-module, there is an f ∈ T which extends h fromN toMc. For any t∈T and n∈N,
(nt)f−(n)f t= (nt)h−(n)ht = 0.
Hence N ⊂ Ke(tf −f t). By (1), N AMc, and since Mc is polyform, tf −f t = 0, implying thatf is an (A, T)-morphism andMc is a self-injective (A, T)-module.
The endomorphism ring of the self-injective (A, T)-module Mc is the centre of the regular ring T and hence is also regular. So Mc is a polyform (A, T)-module by 11.1.
This in turn implies that C is self-injective.
(3) By 11.11, there is a bimodule decomposition Mc = TK(Mc)⊕KTd. Then the projectionε(K) :Mc→KTd is an idempotent in C and
AnC(K) = AnC(KTd) = (1−ε(K))C.
(4) This property is obvious since Mc is polyform.
(5) As shown in 11.11, every cyclic C-submodule of Mc is isomorphic to a direct summand of C and hence is C-injective. Since Mc is a non-singular C-module, any finite sum of C-injective submodules is againC-injective.
(6) Let Mc be generated as (A, T)-module by m1, . . . , mk. Then the map C →Mck, c7→(m1, . . . , mk)c,
is a monomorphism. Since C is injective, it is a direct summand of Mck and so Mc is
a generator in C-Mod. 2
For later use we state some linear dependence properties of elements in Mc with respect toEndA(Mc).
11.13 Independence over the endomorphism ring.
Let M be an A-module, T =EndA(Mc) and m1, . . . , mn∈Mc.
(1) Assume m1 6∈ Pni=2miT. Then there exists a ∈ A such that am1 6= 0 and ami = 0 for i= 2, . . . , n.
(2) Assume and m1T ∩Pni=2miT = 0. Then AnA(m2, . . . , mn)m1Am1.
(3) If M is polyform and AnA(m2, . . . , mn)m1Am1, then m1T ∩Pni=2miT = 0.
Proof. PutU =AnA(m2, . . . , mn).
(1) This follows from the proof of (2).
(2) Assume there exists a non-zero submodule V ⊂Am1 satisfying V ∩U m1 = 0.
Consider the canonical projectionα :V⊕U m1 →V. M being self-injective,αextends to an endomorphismtofM. FromAm1t⊃(V+U m1)t=V 6= 0 we concludem1t6= 0.
To show thatψ is well-defined assumePni=1si(r1mi, . . . , rkmi) = 0.
Then Pni=1sirjmi = 0, for all j = 1, . . . , k.
By assumption, s1 ∈(TU m1(M) :m)A =AnA(me). Hence s1me = 0 proving that ψ is a well-defined morphism.
M being M-injective, ψ can be extended to a morphism Mk →M, also denoted byψ. Since HomA(Mk, M) = Tk there exist t1, . . . , tk∈T such that There is an extension of a module M contained in the M-injective hull Mc which turns out to be of some interest.
Definition. Let M be an A-module, T = EndA(Mc) and B the Boolean ring of all central idempotents of T. Then we callMf=M B the idempotent closure of M.
This notion is closely related to the π-injective hull of M defined in Goel-Jain [143], which can be written asM U, withU the subring generated by all idempotents inT. Hence if all idempotents inT are central, Mf is just the π-injective hull of M. 11.15 Idempotent closure of polyform modules.
We use the above notation. Let M be an A-module with idempotent closure Mf. Then for every a ∈ Mf, there exist m1, . . . , mk ∈ M and pairwise orthogonal to (ZZ2)k for some k ∈IN. So it contains a subset of pairwise orthogonal idempotents c1, . . . , ck such that, for all j ≤r, bj =Pi∈S(j)ci with S(j)⊂ {1, . . . k}. Hence
AssumeM is polyform. Putei =ε(mi)ciε(a). Since aε(a) = aand miε(mi) =mi,
(2) follows from the definition of the idempotents ei above.
(3) Clearly ε(a)ei =ei, for all i= 1, . . . , k, and hence ε(a)Pki=1ei =Pki=1ei.
(2) Show that the following are equivalent for an A-module M: (a) HomA(M/SocM,Mc) = 0;
(b) M is polyform andSocM M;
(c) the canonical assignement EndA(SocM)→EndA(Mc)is an isomorphism.
(3) Strongly regular endomorphism ring ([226]).
Prove that for an A-module M and T =EndA(Mc)the following are equivalent:
(a) T is strongly regular;
(b) M is polyform andT is reduced;
(c) every direct summand in Mc is fully invariant.
(4) Let M be a self-injective polyform A-module and assume every submodule of M contains a uniform submodule. Prove that EndA(M) is a product of full linear rings (endomorphism rings of vector spaces, [200, Theorem 5.12]).
References. Beidar-Wisbauer [75, 76, 77], Dung-Huynh-Smith-Wisbauer [11], Lambek [183], Leu [187], Miyashita [200], Renault [226], Smith [245], Stenstr¨om [39], Wisbauer [273, 279], Zelmanowitz [284, 285].