14 Semiprime modules

1.Lemma. 2.Properties of trace and torsion submodules. 3.Strongly semiprime mod-ules. 4.Characterizations. 5.SSP and semisimple modmod-ules. 6.Properly semiprime modules. 7.Properties of PSP-modules. 8.Corollary. 9.Relation to strongly prime modules. 10.Polyform SSP modules. 11.Polyform PSP modules. 12.Self-injective PSP modules. 13.Self-injective finitely presented PSP modules. 14.Density property of the self-injective hull. 15.Duo endomorphism rings. 16.Projective PSP modules.

17.Projective SSP modules. 18.Torsion submodules of semiprime rings. 19.Left SSP rings. 20.Left PSP rings. 21.Pseudo regular modules. 22.Left fully idempotent rings.

23.Pseudo regular and P SP modules. 24.Remarks. 25.Exercises.

In this section we extend the notion of semiprimeness from (commutative) rings to modules. We refer to 14.24 for a list of various possibilities to do so. In view of our applications we are mainly interested in non-projective modules.

The two following technical lemmas will be crucial for our investigations.

14.1 Lemma. Let M be an A-module and K, L⊂M. The following are equivalent:

(a) M/L∈σ[K];

(b) for any b ∈M, there exists a finite subset X ⊂K, with An_A(X)b⊂L.

Proof. (a) ⇒ (b) Assume M/L ∈ σ[K] and b ∈ M. Then Ab+L/L ⊂ M/L is a cyclic module in σ[K] and hence a factor module of a cyclic submodule of K^(IN). So there existx₁, . . . , x_k∈K and a morphism

A(x1, . . . , xk)→M/L, a(x1, . . . , xk)7→a(b+L)/L.

This impliesAn_A(x₁, . . . , x_k)b ⊂L.

(b) ⇒ (a) Let b ∈ M and choose x₁, . . . , x_k ∈K with An_A(x₁, . . . , x_k)b ⊂ L. We define a map as given above and so (Ab+L)/L∈σ[K]. Hence M/L∈σ[K]. 2

We use the notation for trace and torsion submodules introduced in 11.9.

14.2 Properties of trace and torsion submodules.

Let M be an A-module, K ⊂M andT =End_A(M^c). The following are equivalent:

(a) M/T_K(M)∈σ[K];

(b) for any b ∈M, there exists a finite subset X ⊂K, with An_A(X)b⊂ T_K(M).

(c) everyK-injective,T_K-torsionfree module inσ[M]isM-injective andK-generated;

(d) KT is M-injective and T^K(M) +T_K(M)M;

(e) M^c=KT ⊕I_M(T_K(M^c)), where I_M denotes the M-injective hull.

Notice that the decomposition of M^c given in (e) is in A-Mod. Though KT obvi-ously is a fully invariant submodule this need not be true for I_M(T_K(M^c)).

Proof. (a)⇔(b) follows from 14.1.

(a)⇒(c) AssumeQ∈σ[M] isK-injective andT_K-torsionfree. For any submodule L ⊂ M, consider a morphism f : L → Q. Since TK(Q) = 0, f factorizes through f⁰ :L/T_K(L)→Q. We have the commutative diagram (with canonical mappings)

0 → L → M

↓ ↓

0 → L/T_K(L) → M/T_K(M)

↓_f⁰ Q

Since Q is K-injective and M/T_K(M) ∈ σ[K], there exists some M/T_K(M) → Q yielding a commutative diagram. HenceQ is M-injective and

Q=T r(M, Q) = T r(M/T_K(M), Q) =T r(K, Q).

(c)⇒(a) SinceIM(M/TK(M)) isM-injective andTK-torsionfree, it isK-generated by (c) and soM/T_K(M)∈σ[K].

(a) ⇒ (d) As shown above, I_M(M/T_K(M)) ∈ σ[K]. For a complement U of TK(M) in M, U ⊕ TK(M)M and U is isomorphic to a submodule of M/TK(M).

Hence U ⊂ T^K(M) and T^K(M) +T_K(M)M.

(d)⇒(a) SinceT^K(M) +T_K(M)M and T^K(M)∩ T_K(M) = 0, T^K(M)'[T^K(M) +T_K(M)]/T_K(M)M/T_K(M) . Hence M/T_K(M) is isomorphic to a submodule of KT^d ∈σ[K].

(d)⇒(e) The assumptions implyM^c =KT^d ⊕IM(TK(M)). As an injective object, KTd ∈σ[K] is K-generated and hence KT^d =KT.

Since T^K(M)T^K(M^c), KT^d =I_M(T^K(M^c)).

(e)⇒(d) As a direct summand ofM^c,KT is M-injective. HenceT^K(M)KT^d = KT. SinceTK(M)TK(M^c)IM(TK(M^c)), we concludeT^K(M) +TK(M)M. 2

Referring to the above relations we define:

14.3 Strongly semiprime modules.

Let M be an A-module and T =End_A(M).^c M is called strongly semiprime (SSP) if it satisfies the following equivalent conditions for every submodule K ⊂M:

(a) M/TK(M)∈σ[K];

(b) for any b ∈M, there exists a finite subset X ⊂K, with An_A(X)b⊂ T_K(M);

(c) everyK-injectiveT_K-torsionfree module inσ[M]isM-injective andK-generated;

(d) KT^d ∈σ[K] and T^K(M) +T_K(M)M;

(e) M^c=KT ⊕I_M(T_K(M^c)), where I_M denotes the M-injective hull.

Any SP module is SSP (M^c = KT). Also every semisimple module is SSP. We state some basic important properties.

14.4 Characterizations.

Let M be an A-module, S = End_A(M) and T = End_A(M). Then the following^c are equivalent:

(a) AM is SSP;

(b) for any N AM, M ∈σ[N], and for any K ⊂AM, T^K(M) +TK(M)AM; (c) _AM^c is SSP;

(d) M^c is a semisimple (A, T)-bimodule.

Proof. (a)⇒(b) ForNM,T^N(M)∩T_N(M) = 0 impliesT_N(M) = 0 andN T =M,^c in particularM ∈σ[N].

(b) ⇒ (a) Let K ⊂ M be any submodule. Since T_K(M) +T^K(M)M, M ∈ σ[T_K(M) +T^K(M)] =σ[M] by assumption. SoT_K(M)+Kis a subgenerator inσ[M]

and hence it generates theM-injective moduleKT^d. However,Hom_A(T_K(M),KT^d) = 0 implies thatKT^d is K-generated and M is an SSP module.

(a) ⇒ (c) For any submodule N ⊂ M^c, put K = N T ∩M. Then KT N T and T_K(M^c) = T_N(M^c). Consider any N-injective and T_N-torsionfree module Q ∈ σ[M].

Then Q is K-injective and T_K-torsionfree, and hence M-injective and K-generated since M is SSP (cf. 14.3). As easily seen, Qis also N-generated and soM^c is SSP by 14.3.

(c)⇒(a) Essential submodules of SSP modules are obviously SSP.

(c)⇔(d) Put M =M^c. Let U ⊂M be an essential (A, S)-submodule.

Then U = T^U(M), and T^U(M)∩ T_U(M) = 0 implies T_U(M) = 0. We see from 14.3 that U = M. So M has no proper essential (A, S)-submodule. Hence it is a semisimple (A, S)-module.

Now assumeM is a semisimple (A, S)-module andK ⊂M anA-submodule. Then M =KS ⊕L for some fully invariant L ⊂ M. This implies Hom_A(L, KS) = 0 and soL⊂ T_K(M). Hence M/T_K(M)∈σ[M/L] =σ[K] showing that M is SSP. 2

14.5 SSP and semisimple modules.

Let M be an A-module.

(1) Assume M has essential socle and for every N M, M ∈ σ[N]. Then M is semisimple.

(2) M is semisimple if and only if every module in σ[M] is SSP.

Proof. (1) By assumption,M ∈σ[Soc(M)] and modules inσ[Soc(M)] are semisimple.

(2) We see from (1) that every finitely cogenerated module in σ[M] is semisimple and hence every simple module inσ[M] isM-injective, i.e.,M is co-semisimple.

Let N be the sum of all non-isomorphic simple modules in σ[M] and consider L=M ⊕N. Then T_N(L)⊂Rad(L) = 0 (cf. [40], 23.1). Since L is SSP, this implies L/T_N(L)∈σ[N]. Hence L and M are semisimple modules. 2

Weakening the conditions for strongly semiprime modules we define:

14.6 Properly semiprime modules.

Let M be a left A-module and T = End_A(M^c). We call M properly semiprime (PSP) if it satisfies the following equivalent conditions:

(a) For every element a∈M, M/T_a(M)∈σ[Ra];

(b) for any a, b∈M, there exist r₁, . . . , r_n∈A such that AnA(r1a, r2a, . . . , rna)b ⊂ Ta(M);

(c) for every finitely generated submodule K ⊂M, M/T_K(M)∈σ[K];

(d) for any cyclic K ⊂M, every K-injective T_K-torsionfree module in σ[M] is M-injective and K-generated;

(e) for any cyclic K ⊂M, KT^d ∈σ[K] and T^K(M) +TK(M)M; (f ) for any cyclic K ⊂M, M^c =KT ⊕I_M(T_K(M^c)).

Conditions (d)-(f ) also hold for finitely generated submodules.

Proof. For the equivalence of (a), (c), (d), (e) and (f) see 14.2. (c)⇔(b) is clear.

(b) ⇒ (c) Let a₁, . . . , a_k be a generating subset of K. By 11.10, T_K(M) =

Tk

i=1T_ai(M). Hence

M/T_K(M)⊂

k

M

i=1

M/T_ai(M).

But M/T_ai(M)∈σ[Ra_i]⊂σ[K]. Thus M/T_K(M)∈σ[K]. 2

14.7 Properties of PSP-modules.

Let M be an A-module and T =End_A(M^c).

(1) Assume M is a PSP-module and U ⊂M^c an (A, T)-submodule of finite uniform dimension. Then U is a semisimple (A, T)-bimodule.

(2) Assume M^c has finite uniform dimension as (A, T)-bimodule. Then M is an SSP-module if and only ifM is a PSP-module.

Proof. (1) First assume U is a uniform (A, T)-bimodule. Let V ⊂U be an (A, T )-submodule, and N ⊂ M ∩V a finitely generated A-submodule. Then N T is an essential (A, T)-submodule of U. T^N(U)∩ TN(U) = 0 implies TN(U) = 0.

From this we deduce N T V U as A-modules. Since N T is M-injective, we conclude N T =V =U and U is a simple (A, T)-bimodule.

Now assumeU has finite uniform dimension as (A, T)-bimodule. Hence there exist uniform submodulesV_i ⊂U,i= 1, . . . , n, such that^Ln_i=1V_iU as (A, T)-submodule.

Let N_i be finitely generated submodule of the left A-module V_i ∩ M and N =

Ln

i=1Ni. As shown above, all Vi = NiT are simple (A, T)-bimodules. Hence N T =

Pn

i=1N_iT =^Ln_i=1V_iU as A-submodule. Therefore U =N T =^Ln_i=1V_i is a finitely generated semisimple (A, T)-bimodule.

(2) If M is SSP then obviouslyM is PSP.

If M is PSP and M^c has finite uniform dimension as (A, T)-bimodule, M^c is a semisimple (A, T)-bimodule by (1) and hence M is SSP by 14.4. 2 14.8 Corollary. For a finitely generated left A-module M and T = EndA(M^c), the

following are equivalent:

(a) M is an SSP-module;

(b) M^c is (finitely generated and) semisimple as an (A, T)-bimodule;

(c) M is PSP and M^c has finite uniform dimension as an (A, T)-bimodule.

Proof. Since M is a finitely generated A-module, M^c = M T is a finitely generated (A, T)-bimodule. Hence the assertions follow from 14.4 and 14.7. 2

The next result shows the relation to strongly prime modules.

14.9 Relation to strongly prime modules.

For anA-module M with T =End_A(M^c), the following are equivalent:

(a) M is strongly prime;

(b) M is PSP (or SSP) and M^c is a uniform (A, T)-bimodule.

(c) M^c is a simple (A, T)-bimodule.

In particular, for a uniform A-module M, the conditions strongly prime, SSP, and PSP are equivalent.

Proof. (a)⇒(b) For every submoduleK ⊂M,KT =M^c and the assertion is clear.

(b) ⇒ (a) Assume M is PSP and K ⊂ M is a finitely generated submodule. By the uniformity condition, T^K(M^c)∩ T_K(M^c) = 0 implies T_K(M^c) = 0 and M^c = KT. SoM is strongly prime.

(a)⇔(c) This follows with the same arguments as applied in the proof of 14.4. 2 In general, SSP modules need not be polyform. Modules satisfying both conditions have particularly nice structure properties.

14.10 Polyform SSP modules.

For a polyform A-module M and T =End_A(M^c), the following are equivalent:

(a) M is an SSP-module;

(b) for every N M, M ∈σ[N];

(c) for every submodule K ⊂M, KT^c ∈σ[K];

(d) for every submodule K ⊂M, M^c =KT ⊕ T_K(M^c).

Proof. The statements follow from 11.11, 14.3 and 14.4. 2 We know that a module M is SSP if and only if M^c is SSP. In general, for a PSP module M,M^c need not be PSP. For polyform modules the PSP property extends at least to the idempotent closure:

14.11 Polyform PSP modules.

Let M be a polyform A-module, T = End_A(M^c), B the Boolean ring of central idempotents of T, and M^f the idempotent closure of M. Then the following are equiv-alent:

(a) M is a PSP module;

(b) for every m∈M, AmT =M ε(m);^c

(c) for every finitely generated submodule K ⊂M, KT =M ε(K);^c (d) M^f is a PSP module.

Under the given conditions, T_m(M^c) =M^c(id−ε(m)).

Proof. (a) ⇒ (b) By 14.6(f) and 11.11(1), M^c = AmT ⊕ T_m(M^c). Now (b) follows from the definition of the idempotentε(m).

(b)⇒(a) Since M ε(m) is^c M-injective,

Pierce stalks of polyform modules will be considered in 18.16.

By 14.4, any injective SSP module is semisimple as a bimodule. For self-injective polyform PSP modules we get a weaker structure theorem:

14.12 Self-injective PSP modules.

Let M be a self-injective polyform A-module and T = EndA(M^c). Denote Λ = A⊗_ZZT^o and C =End_Λ(M). Then the following conditions are equivalent:

(a) _AM is a PSP A-module;

(b) every cyclic Λ-submodule of M is a direct summand;

(c) every finitely generated Λ-submodule of M is a direct summand;

(d) as a Λ-module, M is a selfgenerator;

(e) for any m ∈M and f ∈EndC(M), there exists h ∈Λ with f(m) =hm.

Proof. Notice that Λ-submodules of M are just fully invariant submodules and C can be identified with the centre of T. HenceC is a commutative regular ring.

(a)⇔(b)⇔(c) is clear by 14.11, (c)⇒(d) is obvious.

(d)⇒(e) This follows from the proof of the Density Theorem (e.g., [40, 15.7]).

(e) ⇒ (b) Choose any m ∈ M and ε(m) ∈ C as defined in 11.12. Since mC ' ε(m)C is a direct summand, for any n ∈M, there exists f ∈End_C(M) with f(m) = nε(m). By (e),f(m) =hm for some h∈Λ and hence M ε(m)⊂Λm.

On the other hand, m = mε(m) and Λm ⊂ (ΛM)ε(m) ⊂ M ε(m). So Λm = M ε(m) is a direct summand and the assertion is proved. 2

ModulesM, whose finitely generated submodules are direct summands, are closely related to M being regular in σ[M]. In fact, if M is finitely presented inσ[M], these two notions coincide (7.3). As a special case we derive from above:

14.13 Self-injective finitely presented PSP modules.

Let M be a self-injective polyform A-module and T = End_A(M^c). Denote Λ = A⊗_ZZ T^o and C =End_Λ(M). Assume M is finitely generated as a Λ-module. Then the following conditions are equivalent:

(a) M is a PSP A-module and finitely presented in σ[_ΛM];

(b) ΛM is regular and projective in σ[ΛM];

(c) _ΛM is a (projective) generator in σ[_ΛM];

(d) for any m₁, . . . , m_n ∈M and f ∈End_C(M), there exists h∈Λ with f(m_i) = hm_i, for i= 1, . . . , n (density property).

Proof. Since _ΛM is finitely generated, M_C is a generator in C-Modby 11.12. So M_C is a faithfully flat C-module.

(a) ⇔ (b) By 14.12, every finitely generated Λ-submodule of M is a direct sum-mand. Now the assertion follows from 7.3.

(b)⇒(c)⇒(d) are obvious (Density Theorem 5.4).

(d) ⇒ (a) Since M_C is a generator in C-Mod, M is (finitely generated and) pro-jective as an End_C(M)-module (5.5).

By the density property, the categories σ[ΛM] and σ[EndC(M)M] coincide (5.4).

SoM is projective (hence finitely presented) in σ[_ΛM].

By 14.12, M is a PSP A-module. 2

For modules M with End_A(M^c) commutative we are now able to characterize the density property of M^c as bimodule.

14.14 Density property of the self-injective hull.

Let M be a polyformA-module, assume T =End_A(M^c) to be commutative and put Λ =A⊗_ZZT. Then the following are equivalent:

(a) For any a₁, . . . , a_n∈M^c and f ∈End_T(M^c), there exists h∈Λ with hai =f ai for i= 1, . . . , n.

(b) For any m1, . . . , mn, m∈M there exist r1, . . . , rk∈A such that for s₁, . . . , s_n∈A, the relations

n

X

l=1

s_lr_jm_l= 0 for j = 1, . . . , k, imply s1m∈ TU m1(M) for U =AnA(m2, . . . , mn).

(c) M is a PSP module and for any m₁, . . . , m_n ∈ M, there exist r₁, . . . , r_k ∈ A such that fors₁, . . . , s_n ∈A the relations

n

X

l=1

s_lr_jm_l= 0 for j = 1, . . . , k, imply s₁m₁ ∈ T_{U m1}(M) for U =An_A(m₂, . . . , m_n).

If M^c is finitely presented in σ[_ΛM^c], the above are equivalent to:

(d) M^c is a PSP A-module.

Proof. PutU =An_A(m₂, . . . , m_n).

(a)⇒(b) PutN =^Pn_i=1miT. By 11.12,M^cis a non-singular andN is an injective T-module. Since in a non-singular module the intersection of injective submodules is again injective,K =m₁T∩N isT-injective. Hencem₁T =K⊕Lfor some submodule L⊂m₁T. By 11.12, An_T(m₁) = (1−ε(m₁))T. This means that the map

m₁T →ε(m₁)T, m₁t7→ε(m₁)t,

is an isomorphism. So there exist idempotents u, v ∈T with the properties (∗) uv = 0, u+v =ε(m1), K =m1uT 'uT and L=m1vT 'vT.

Since m₁uT =K ⊂N and U N = 0, we have U m₁u= 0 and U m₁ =U[m₁ε(m₁)] = U[m₁(u+v)] =U m₁v.

By 11.13, U m₁vAm₁v, and by 11.12, ε(U m₁v) =ε(Am₁v) =ε(m₁v).

The isomorphismm₁vT 'vT implies ε(m₁v) =v and hence we have (∗∗) ε(U m₁) =v .

Since m∈M and T_{U m1}(M) = M ∩ T_{U m1}(M^c), by 11.14,

(TU m1(M) :m)A = (TU m1(M^c) :m)A =AnA(mv).

As an injective submodule,m₁vT ⊕N is a direct summand inM^c. Hence there exists a T-endomorphism ψ of M^c satisfying ψN = 0 and ψm₁v =mv (recall m₁vT ' vT).

Since m1u∈N, we have ψm1u= 0 and ψm1 =ψ[m(u+v)] =mv.

By assumption, there exists h∈Λ satisfying

hm1 =ψm1 =mv, and hmi =ψmi for i= 2, . . . , n . Now the assertion follows from 11.14.

(b) ⇒ (c) Putting m₁ = · · · = m_n = 0 we see that M is a PSP module. The second part of the conditions in (c) follows from (b) form =m1.

(c) ⇒ (a) Since M^c = M T it suffices to show that for any m₁, . . . , m_n ∈ M and f ∈End_T(M^c), there existsh ∈Λ such thatf m_i =hm_i, for all i= 1, . . . , n.

We prove this by induction on the cardinality|I|of minimal subsetsI ⊂ {1, . . . , n}

satisfying ^P_i∈ImiT =^Pn_i=1miT.

Consider the case |I|= 1, i.e., I ={1}. By 14.11, M ε(m^c ₁) = Λm₁. Since f m₁ =f[m₁ε(m₁)] = (f m₁)ε(m₁)∈M ε(m^c ₁) = Λm₁ ,

f m₁ =hm₁ for someh∈Λ. By assumption, ^Pn_i=1m_iT =m₁T, implyingf m_i =hm_i, for all i= 1, . . . , n.

Now assume |I| = n and consider N = ^Pn_i=2m_iT. As shown above, there exist idempotentsu, v ∈T satisfying (∗) and (∗∗). By 11.14, there exists h₁ ∈Λ with

h₁m₁ =m₁v and h₁m_i = 0, for i= 2, . . . , n . Notice that h₁m₁u= (h₁m₁)u= (m₁v)u=m₁(vu) = 0.

By induction hypothesis, there existsh₂ ∈Λ with h₂m_i =f m_i, fori= 2, . . . , n .

Obviously, h₂x=f xfor any x∈N. From (∗) we obtain h₂m₁u=f m₁u.

Consider the elementm =f m₁v−h₂m₁v. Clearly m=mv and so m ∈M v. By (∗),^c vε(m1) = v. Now it follows from 14.11 that

Λm₁v = (Λm₁)v =M ε(m^c ₁)v =M v .^c Hence there existsh₃ ∈Λ with

h₃m₁v =m=f m₁v−h₂m₁v . Puttingh=h3h1 +h2, we have

hm₁ =h₃(h₁m₁u) +h₃(h₁m₁v) +h₂m₁u+h₂m₁v

=h₃m₁v+f m₁u+h₂m₁v

= (f m₁v−h₂m₁v) +f m₁u+h₂m₁v

=f m1v+f m1u=f m1 ,

hm_i =h₃(h₁m_i) +h₂m_i =f m_i, for i= 2, . . . , n .

(a)⇔(d) This is clear by 14.13. 2

We have seen in 14.12 and 14.13 that self-injective polyform PSP modules have nice structural properties. Hence we may ask for which modulesM, the M-injective hullM^c is a PSP module.

Now we turn to the question which additional conditons on an SSP or PSP module M imply that M is polyform. It is interesting to observe that this is achieved by commutativity conditions on the endomorphism ring as well as by projectivity of the module. Recall that a ring is said to be (left and right) duo if all its one-sided ideals are two-sided.

14.15 Duo endomorphism rings.

Let M be an A-module, T =End_A(M^c) and assume, for every NM, M ∈σ[N].

(1) Jac(T)∩Z(T) = 0.

(2) Suppose T is a duo ring. Then M is polyform and SSP.

Proof. (2) Assume for f ∈T, N =KefM. Then^c M ∈ σ[N] which is equivalent to N T = M^c. This implies M f^c = (N T)f = (N f)T = 0 and hence f = 0. So the Jacobson radical of T is zero, i.e., M is polyform. By 14.10,M is SSP.

(1) A similar argument also implies this assertion. 2 Projectivity makes any PSP module polyform. This applies in particular for the left module structure of the ring itself.

14.16 Projective PSP modules.

Let M be a PSP module which is projective in σ[M]. Then:

(1) For any submodules K, N ⊂M, T_N(K)is a complement ofT^N(K)in K, hence T^N(K) +T_N(K)K and [T^N(K) +T_N(K)]/T_N(K)K/T_N(K).

(2) M is polyform.

Proof. M is projective in σ[M] if and only if M^(Λ) is self-projective, for any set Λ.

From this it is obvious that M/X is projective in σ[M/X], for every fully invariant submoduleX ⊂M.

(1) First we show T^N(K) 6= 0, for any finitely generated submodules K, N ⊂M with Hom(K, N) 6= 0. We may assume that there is an epimorphism f : K → N.

Then N ⊂ T^K(M) and

T_K(M)∩N = 0 =T_K(M)∩K.

PutL=T_K(M) andM =M/L. There are canonical inclusionsN ⊂M andK ⊂M.

Since M is PSP, M ∈σ[K] and so σ[M] =σ[K].

As outlined above, M is projective in σ[K] and hence is a submodule of K^(Λ), for some set Λ. Therefore the compostion of the inclusion N ⊂ M with a suitable map M →K yields a non-zero morphism N →K. This meansT^N(K)6= 0.

From 11.10 we know T^N(K) ∩ T_N(K) = 0. We have to show that T_N(K) is maximal with respect to this property.

For x ∈ K \ T_N(K), there exists a non-zero g : Ax → N^cand 0 6= (y)g ∈ N, for some y∈Ax. From the above we know

06=T_N(Ay)⊂ T^N(K)∩Ax.

This shows the maximality of T_N(K) which implies the assertions (see [40, 140]).

(2) AssumeM is not polyform. Then there exist a cyclic submoduleK ⊂M and a non-zero morphismf :K →M withKefK. Put N = (K)f and M =M/T_N(M).

The map K →^f M →M factorizes through ¯f :K/T_N(K)→M. Since T^N(K)∩Kef T^N(K) and T^N(K)K/T_N(K) by (1), we observeKef¯K/TN(K) and hence N 'Imf¯is an M-singular module.

By assumption, M ∈σ[N] =σ[M]. So, in particular, M is M-singular. However, as noted above, M is projective in σ[M] and hence cannot be M-singular.

Therefore M is polyform. 2

14.17 Projective SSP modules.

LetM be projective inσ[M]andT =End_A(M^c). Then the following are equivalent:

(a) M is an SSP-module;

(b) for every submodule K ⊂M, M^c =KT ⊕ TK(M^c).

(c) M is polyform and for any N M, M ∈σ[N].

Proof. (a)⇒(b) By 14.16, M is polyform and now apply 14.10.

(b) ⇒(a) The decomposition implies in particular that every fully invariant sub-module is a direct summand in M^c as (A, T)-submodule. Hence M^c is a semisimple (A, T)-bimodule and M is SSP by 14.4.

(a)⇔(c) By 14.16, M is polyform and the assertion follows from 14.10. 2 Definition. A ring A is calledleft PSP (SSP), if _AA is a PSP (SSP) module.

Notice that the definition also applies to rings without units, considering such rings as modules over rings with units in a canonical way. Rings with units are (left) projective and hence are left non-singular if they are left PSP (or SSP) (by 14.16). We will see that they are also semiprime. Before we want to describe the torsion modules related to semiprime rings.

14.18 Torsion submodules of semiprime rings.

Let A be a semiprime ring and N ⊂A a left ideal. Then:

(1) T_N(A) = An_A(N).

(2) A/T_N(A)∈σ[N]if and only if there exists a finite subsetX ⊂N withAn_A(X) = An_A(N).

Proof. (1) The relation T_N(A) ⊂ An_A(N) always holds. Clearly N An_A(N) is a nilpotent left ideal and hence is zero.

Consider f ∈Hom_A(An_A(N), I_A(N)) and put K = (N)f⁻¹. Then (Kf)² ⊂N(Kf) = (N K)f ⊂(N An_A(N))f = 0.

Since A is semiprime, Kf = 0 and so Im f ∩N = 0, implying f = 0. Hence TN(A)⊃AnA(N).

(2) Assume A/T_N(A)∈σ[N]. By 14.1, there exists a finite subset X ⊂N with AnA(X)1⊂ TN(A) = AnA(N)⊂AnA(X).

Now assume An_A(X) =An_A(N) for some finite X ⊂N. Then An_A(X) =T_N(A) by (1), and for any b ∈A,An_A(X)b⊂ T_N(A). Now apply 14.1. 2

Applying our module theoretic results to _AA, we obtain characterizations of 14.19 Left SSP rings.

For the ring A put Q:=Q_max(A). The following are equivalent:

(a) A is left SSP;

(b) for every essential left ideal N ⊂A, A∈σ[N];

(c) every essential left ideal N ⊂A contains a finite subset X with An_A(X) = 0;

(d) for every left ideal I ⊂A, Q=IQ⊕ T_I(Q);

(e) A is semiprime and every left ideal I ⊂ A contains a finite subset X ⊂ I with An_A(X) = An_A(I);

(f ) Q is a semisimple (A, Q)-module.

If A satisfies these conditions, then Q is left self-injective, von Neumann regular, and a finite product of simple rings.

Proof. (a)⇒(b) is shown in 14.4.

(b) ⇒(a) Assume for every essential left ideal N ⊂ A, A∈ σ[N]. Any such N is a faithful A-module.

First we show that A is semiprime. For this consider an ideal I ⊂A with I² = 0.

Let J be the right annihilator of I, and L ⊂ A any non-zero left ideal. Obviously, I ⊂J. Assume L∩J = 0. Then IL6= 0. However, IL⊂L∩J = 0, a contradiction.

This implies that J is an essential left ideal in A. By our assumption, J is a faithful left module and IJ = 0 means I = 0.

In view of 14.4 and 11.11, it remains to show thatA is left non-singular

If the left singular ideal S(A)⊂ A is non-zero, S(A)⊕AnA(S(A)) is an essential left ideal inA. Hence there area₁, . . . , a_k withAn_A(a₁, . . . , a_k) = 0. From this we see that there is a monomorphism S(A)→ S(A)(b₁, . . . , b_r), withb₁, . . . , b_r ∈ S(A).

The kernel of this map isS(A)∩An_A(b₁)∩. . .∩An_A(b_r). Since all the An_A(b_i) are essential left ideals in A, this intersection could not be zero, a contradiction. Hence A is left non-singular.

(b)⇔(c) This is obvious by 14.1.

(a) ⇒(d) By 14.16, A is semiprime and left non-singular. Therefore Q = A^b and End_A(Q) = Q. Now the assertion follows from 14.10.

(d) ⇒ (a) We show that A is left non-singular. Assume for a ∈ A, AnA(a) is an essential left ideal in A. Since Q = RaQ⊕ T_a(Q), we have RaQ = eQ, for some idempotente ∈Q. Write e=^Pn_i=1r_iaq_i, withr_i ∈A, q_i ∈Q.

Obviously, L = ^Tn_i=1(An_A(a) : r_i)_A is an essential left ideal in A and Le = 0.

ThereforeL∩Qe= 0 and also L∩(Qe∩A) = 0. However,L⊂A is an essential left ideal and Qe∩A 6= 0, a contradiction. Hence A is left non-singular and so Q = A^b and End_A(Q) =Q. Now apply 14.10.

(a)⇔(e)⇔(f) follow from 14.17, 14.3 and 14.4. 2 14.20 Left PSP rings.

For a ring A, the following conditions are equivalent:

(a) A is a left PSP-ring;

(b) Ais semiprime (and left non-singular) and for every finitely generated left ideal N ⊂A, An_A(N) =An_A(X), for some finite subset X ⊂N.

Proof. (a) ⇒ (b) By 14.16, A is left non-singular. To show that A is semiprime, consider a cyclic left ideal N ∈ A with N² = 0. By assumption, A/T_N(A) ∈ σ[N].

Since N K = 0 for any K ∈σ[N], in particular N(A/T_N(A)) = 0 and N A ⊂ T_N(A).

This impliesN A ⊂ T^N(A)∩ T_N(A) = 0 and N = 0. Now refer to 14.18.

(b)⇒(a) also follows from 14.18. 2

Now we consider a condition on modules which is closely related to regularity properties.

14.21 Pseudo regular modules.

Let M be a left A-module, S = EndA(M), T = EndA(M^c) and D = EndT(M^c).

Then the following conditions are equivalent:

(a) For every m∈M, there exists h∈D, such that hM ⊂AmS and hm =m;

(b) for any (A, S)-submodule N ⊂ M and m₁, . . . , m_k ∈ N, there exists h ∈ D, such that hM ⊂N and hm_i =m_i, for all i= 1, . . . , k.

A module satisfying these conditions is called pseudo regular.

Proof. (a) ⇒ (b) Let N ⊂ M be an (A, S)-submodule of M and m₁, . . . , m_k ∈ N.

Fork = 1 there is nothing to show.

Assume the assertion holds for anyk−1 elements inN. Choose f ∈D such that f m_k=m_k and f M ⊂Am_kS⊂N.

Clearly m_i −f m_i ∈ N, for all i = 1, . . . , k −1, and by hypothesis, there exists g ∈D with gM ⊂N and

g(m_i −f m_i) = m_i−f m_i, for i= 1, . . . , k−1.

Puth= 1−(1−g)(1−f). Then hm =gm−gf m+f m∈N, for allm ∈M, which means hM ⊂N, and for i= 1, . . . , k,

hm_i =m_i−(1−g)(1−f)m_i =m_i−(1−g)(m_i−f m_i) =m_i.

(b)⇒(a) is obvious. 2

The next result shows what pseudo regularity means for rings.

14.22 Left fully idempotent rings.

A ring A is left fully idempotent if and only if the left A-module A is pseudo regular.

Proof. RecallEnd_A(A) =A. AssumeAis left fully idempotent. LetN be an ideal of A andm ∈N. Since (Am)² =Am, there exist h∈AmAsuch thathm =m. Clearly hA⊂N and h∈A⊂Biend_A(A). Hence^b _AA is pseudo regular.

Now assume_AAto be pseudo regular andm∈A. Then there existsh∈Biend_A(A)^b such that hm=m andhA ⊂AmA. Hence r=h1∈AmA andrm= (h1)m =hm = m. So m =rm∈(Am)², (Am)² =Amand the ring A is left fully idempotent. 2 It follows from the above observation that a commutative ring A is von Neumann regular if and only if_AA is pseudo regular.

Of particular interest for our investigations is the following relationship:

14.23 Pseudo regular and PSP modules.

Let M be a left A-module, S = End_A(M) and T = End_A(M^c). The following are equivalent:

(a) For every m∈M, there is a central idempotent e∈T, with AmS =M e;

(b) for every m∈M, M =AmS ⊕ T_m(M) and M^c=AmT ⊕ T_m(M^c);

(c) for every finitely generated submodule N ⊂ M, there is a central idempotent e∈T, with N S =M e;

(d) for every finitely generated submodule N ⊂M, M =N S⊕ TN(M) and Mc=N T ⊕ T_N(M^c);

(e) M is a pseudo regular P SP-module over A.

Proof. Denote D=EndT(M^c).

(a)⇒(b) Form ∈M choose a central idempotente∈T, with AmS =M e. Since Mc =M T we have

Mc = M T e⊕M T(1−e) =M eT ⊕M T(1−e)

= AmST ⊕M T(1−e) = AmT ⊕M T(1−e).

Hence AmT is M-injective and M T e = AmT = Tm(M^c) = IM(Tm(M^c)) (see 11.10).

Since M^c is M-injective,

Hom_A(M T(1−e), I_M(T_m(M^c))) = {t∈T |M T(1−e)t⊂I_M(T_m(M^c))}

= {t∈T |M T(1−e)t⊂M T e}= 0.

SoM T(1−e)⊂ Tm(M^c). ButM T e=AmT,M^c=M T e⊕M T(1−e) and M T e∩ T_m(M^c) = T^m(M^c)∩ T_m(M^c) = 0.

HenceT_m(M^c) =M T(1−e) andM^c =AmT⊕T_m(M^c). ClearlyAmS =M e=M∩M T e and

T_m(M) =M ∩ T_m(M^c) =M ∩M T(1−e) =M(1−e).

SoM =AmT ⊕ T_m(M).

(b) ⇒ (a) Let e be the projection of M^c onto AmT along T_m(M^c). Since both submodules are fully invariant, e is a central idempotent of T. From AmS ⊂ AmT and T_m(M)⊂ T_m(M^c), we concludeM e=AmS.

(c)⇔(d) can be shown with the above proof.

(c)⇒(a) is obvious.

(a)⇒(e) SinceM/Tm(M) = AmS ∈σ[Am], M isP SP.

Let π be the projection of M^c onto AmT along T_m(M^c). Obviously, π ∈D. Since

Consider the following properties of an A-module M:

(i) A/An_A(M) is cogenerated by every essential submodule of M; (ii) for every N M,M ∈σ[N];

(iii) for every N M,M ⊂N^(Λ), for some set Λ;

(iv) for every N M,A/An_A(M)⊂N^r, for some r∈IN; (v) M is cogenerated by every essential submodule of M; (vi) M is polyform;

(vii) for every submodule K ⊂M, M/T_K(M)∈σ[K];

(viii) for every cyclic submoduleK ⊂M, M/T_K(M)∈σ[K].

The conditions (i)-(vi) stated for every submodule were considered in 13.9. (vii) is used to define strongly semiprime modules, and (viii) defines properly semiprime

modules. For any module M, (iii) ⇒ (v) ⇒ (i), (iii) ⇒ (ii) ⇒ (i), (iv) ⇒ (i) and (vii)⇒(ii).

ForM projective inσ[M], (ii)⇔(iii) and (vii) is equivalent toM satisfying both (ii) and (vi) (cf. 14.16, 14.17).

For A commutative and M = A, any of the properties (i), (v), (vi) and (viii) characterize A as a semiprime ring (hence Qmax(A) is regular). Properties (ii), (iii), (iv) and (vii) imply that Q_max(A) is a finite product of fields.

For M =_AA, the conditions (ii), (iii), (iv) and (vii) are equivalent and describe SSP rings (see 14.19). Left ideals N ⊂ A, which contain a finite subset X with An_A(X) = 0, are also called insulated. Hence A is left SSP if and only if every essential left ideal is insulated. So A is left strongly semiprime ring in the sense of Handelman [149, Theorem 1]. Such rings are also investigated in Kutami-Oshiro [182]

and generalize left strongly prime rings (see 13.9).

14.25 Exercises.

Recall that A is an associative algebra with unit.

(1) The definitions of SSP and PSP modulesM refer to the surrounding category.

Nevertheless submodules of such modules are of the same type.

LetM be anA-module and L⊂M a submodule. Prove: If M is SSP (resp. PSP) (in σ[M]), then L is also SSP (resp. PSP) (in σ[L]).

(2) An A-module M is called weakly semisimple if it is polyform, every finitely generated submodule has finite uniform dimension, and for every non-zero submodule N ⊂M, there exists f ∈Hom_A(M, N) with f|_N 6= 0 (weakly compressible).

Let M be such a module. Prove ([285, 1.1]):

(i) For any m∈M and N M, there exists a monomorphismAm→N ∩Am.

(ii) M is strongly semiprime.

(3) Show that for a ring A, the following are equivalent ([285, 5.1, 5.2]):

(a) A has a faithful weakly semisimple left ideal (of finite uniform dimension);

(b) A is semiprime, left non-singular, with a faithful left ideal which contains a (finite) direct sum of uniform left ideals.

(4) A ring is said to beleft fully idempotentif every left ideal is idempotent. Show that for a ringA, the following are equivalent ([77, 4.2]):

(a) A is biregular;

(b) A is a left fully idempotent left PSP ring;

(c) A is a left PSP ring whose prime factor rings are left fully idempotent;

(d) A is a left PSP ring with all prime ideals maximal.

(5) Let A be a reduced ring. Prove:

(i) For any subset U ⊂A, the left annihilator An_A(U) of U coincides with the right annihilator of U in A ([2, p. 286]).

(ii) A is left (and right) PSP.

(6) Let Abe a reduced ring. Prove that the following are equivalent (see [77, 4.2], [81, Theorem 8]):

(a) A is biregular;

(b) A is left fully idempotent;

(c) all prime ideals in A are maximal.

(7) Let A be a reduced ring. Prove that the following are equivalent ([226]):

(a) Q_max(A) is reduced;

(b) closed left ideals in A are two-sided ideals;

(c) for non-zero elements x, y ∈A, Ax∩Ay= 0 implies xy= 0 =yx.

(8) Let A be a semiprime left non-singular ring. Prove that the following are equivalent ([182, Proposition 2.3]):

(a) Q_max(A) is a direct sum of simple rings;

(b) the set of central idempotents ofQ_max(A)is finite;

(c) A contains no infinite direct sum of ideals.

(9) Show that for a ring A, the following are equivalent ([182, Theorem 2.5]):

(a) A is left SSP;

(b) Qmax(A) is a direct sum of simple rings and for every idempotent e∈Qmax(A), AeQ_max(A) =Q_max(A)eQ_max(A).

(10) Show that for a ring A, the following are equivalent ([182, Theorem 3.4]):

(a) A/S²(A) is left SSP;

(b) every non-singular self-injective leftA-module is injective;

(c) any finite direct sum of non-singular self-injective leftA-modules is self-injective.

(11) Let A be a semiprime ring. Prove ([149, Lemma 7, Corollary 23]):

(i) IfAhas no infinite direct sums of ideals, thenAsatisfies acc and dcc on annihilators of ideals.

(ii) If A has dcc on left annihilators then A is left SSP.

References. Beidar-Wisbauer [75, 76, 77], Birkenmeier-Kim-Park [81], Faith [12], Handelman [149], Kutami-Oshiro [182], Wisbauer [273], Zelmanowitz [285].

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