1.Prime modules with (∗). 2.Proposition. 3.Strongly prime modules. 4.Projective strongly prime modules. 5.Characterization of strongly prime modules. 6.Characteri-zation of left strongly prime rings. 7.Lemma. 8.Strongly prime modules with uniform submodules. 9.Remarks. 10.Exercises.
In this section we consider various conditions on modules M which, for M = A and A commutative, are equivalent to A being a prime ring.
Definitions. An A-module M is called a prime module if A/AnA(M) is cogen-erated by every 0 6= K ⊂ M. M is called strongly prime if for every 0 6= K ⊂ M, M ∈σ[K].
For a prime module M,A/AnA(M) is a prime ring. The ringA is a prime ring if and only if there exists a faithful (or cofaithful) prime (left)A-module (e.g., AA).
There is a property of a module M which allows further conclusions from our primeness conditions:
(∗) For any non-zero submodule K ⊂ M, AnA(M/K) 6⊂ AnA(M), i.e., there is an r∈A\AnA(M) with rM ⊂K.
In general, this need not hold for AA if A is not commutative. However, it will hold for our applications to A as a bimodule.
13.1 Prime modules with (∗).
Let M be an A-module satisfying (∗), S =EndA(M) and A=A/AnA(M).
(1) AM is prime if and only if A is a prime ring.
(2) If AM is prime, then
(i) MS is prime (and S is a prime ring);
(ii) AM is polyform;
(iii) A ∈σ[M] if and only if McT is finitely generated.
Proof. (1) It was already mentioned thatAM prime implies A prime.
Assume Ato be a prime ring andK a nonzero submodule ofAM. By (∗), there is a nonzero ideal I ⊂A with IM ⊂K. For J =AnA(K) we have J I ·M ⊂J ·K = 0, hence J I = 0 andJ = 0.
(2) Let AM be prime. (i) AssumeU t= 0 for anS-submodule U ofMS andt ∈S.
ForI =AnA(M/AU) we get I·M t⊂AU t= 0, implying M t= 0 and t= 0.
(ii) Consider submodules K, L⊂AM with K ⊂L and KM.
ForI =AnA(M/K) andα ∈HomA(L/K, M) we haveI((L/K)α∩K) = 0, hence (L/K)α∩K = 0, implying (L/K)α= 0 and α= 0. So K is rational inM.
(iii) Apply 10.7. 2
13.2 Proposition. For an A-module M satisfying (∗), the following are equivalent:
(a) AM is prime and HomA(M, K)6= 0, for any nonzero submodule K ⊂M; (b) AM is cogenerated by any nonzero submodule K ⊂M.
Proof. (b)⇒(a) is evident.
(a)⇒(b) By 13.1, S =EndA(M) is prime. Hence, for non-zeroK, L⊂M, we get L·HomA(M, K)⊃M ·HomA(M, L)·HomA(M, K)6= 0,
which characterizes (b). 2
We now turn to the investigation of the stronger primeness condition.
13.3 Strongly prime modules.
For an A-module M with M-injective hull Mc, the following are equivalent:
(a) M is strongly prime;
(b) Mc is strongly prime;
(c) Mc is generated by each of its nonzero submodules;
(d) M is contained in every non-zero fully invariant submodule of Mc; (e) Mc has no non-trivial fully invariant submodules;
(f ) for any pretorsion class T in σ[M], T(M) = 0 or T(M) =M; (g) for each m∈M and 06=b∈M, there exist r1, . . . , rk∈A such that
AnA(r1b, . . . , rkb)⊂AnA(m);
(h) there exists a module K ⊃M such that for all 06=m ∈M, M ⊂AmEndA(K).
Proof. (a)⇔ (b) ⇔ (c) This is obvious since Mc ∈ σ[M] and injectives in σ[M] are generated by any subgenerator.
(a)⇒(d) For any fully invariant L⊂M,c M ⊂(L∩M)EndA(Mc)⊂L.
(d)⇒(e) LetL⊂Mcbe fully invariant and M ⊂L. Then Mc =MEndA(Mc)⊂L.
(e)⇒(f) T(Mc) is fully invariant inMc.
(f) ⇒ (b) For every 0 6= K ⊂ Mc, T := σ[K] is a hereditary pretorsion class in σ[M] and T(Mc)6= 0.
(a)⇔(g) The condition given in (g) is equivalent to the existence of a map (Ab)k ⊃A(r1b, . . . , rkb)→Am, a(r1b, . . . , rkb)7→am.
This is equivalent toAm∈σ[Ab].
(c)⇒(h) The module Mc ⊃M has the properties stated.
(h)⇒(a) ClearlyM ∈σ[Am] for all 06=m∈M. 2
It follows from 13.3 that a strongly prime module M is either singular or non-singular inσ[M]. Since projective modules never are singular, any projective strongly prime module in σ[M] is non-M-singular.
13.4 Projective strongly prime modules.
Let M be an A-module which is projective in σ[M]. Assume M is strongly prime.
Then M is polyform and
(1) S=EndA(M) is a left strongly prime ring;
(2) T =EndA(Mc) is the maximal left ring of quotients of S.
Proof. By the preceding remarks, M is polyform (non-M-singular).
(1) Because of projectivity of M, for any left ideal J ⊂ S, J = HomA(M, M J).
Mc being generated by M J, there exist a submodule L ⊂ M J(Λ), Λ some index set, and exact sequences
L→M →0, HomA(M, L)→HomA(M, M)→0.
Hence
S ⊂HomA(M, L)⊂HomA(M, M J)(Λ)=J(Λ), showing that S is left strongly prime.
(2) For any non-zeroK ⊂M,M ∈σ[K]. SinceM is projective inσ[K],M ⊂K(Λ), for some set Λ. In particular, HomA(M, K)6= 0 and 11.5 applies. 2
To get more characterizations of strongly prime modules we introduce a new Definition. A module N ∈ σ[M] is called an absolute subgenerator in σ[M] if every non-zero submodule K ⊂N is a subgenerator in σ[M] (i.e., M ∈σ[K]).
It is obvious that every absolute subgenerator is a strongly prime module, andM itself is an absolute subgenerator inσ[M] if and only if it is strongly prime. Moreover, if N is an absolute subgenerator inσ[M] withSM(N)6= 0, then all modules in σ[M]
are M-singular.
With this notions we have the following
13.5 Characterization of strongly prime modules.
For a polyform A-module M, the following assertions are equivalent;
(a) M is strongly prime;
(b) every module K ∈σ[M] with SM(K)6=K is a subgenerator;
(c) every non-zero module K ∈σ[M] with SM(K) = 0 is an absolute subgenerator;
(d) for every non-zero N ∈σ[M],
SM(N) = \{K ⊂N | N/K is an absolute subgenerator in σ[M]};
(e) there exists an absolute subgenerator in σ[M].
In this case, every projective module in P ∈ σ[M] is an absolute subgenerator and hence SM(P) = 0.
Proof. (a) ⇒ (b) Since M is polyform, the M-singular modules X ∈ σ[M] are characterized by the propertyHomA(X,M) = 0 (see 10.2). Hence for anyc K ∈σ[M]
withSM(K)6=K, there exists a non-zero homomorphism f :K →Mc. Since M (and Mc) is strongly prime, the image off is a subgenerator in σ[M] and so is K.
(b)⇒(c) If SM(N) = 0, every non-zero submodule ofN is non-M-singular.
(c) ⇒ (d) Let K ⊂ N be such that N/K is an absolute subgenerator in σ[M].
Assume SM(N) 6⊂ K. Then (K+SM(N))/K is an M-singular submodule of N/K.
This is impossible since not all modules inσ[M] areM-singular. So SM(N)⊂K and SM(N) is contained in the given intersection.
SinceM is polyform,N/SM(N) is non-M-singular and hence an absolute subgen-erator in σ[M]. This proves our assertion.
(d)⇒(e) SinceSM(M) = 0 the equality in (d) implies the existence of an absolute subgenerator in σ[M].
(e) ⇒(a) Let N be an absolute subgenerator in σ[M]. Then clearly SM(N) = 0 and N is a strongly prime module. Now the proof (a) ⇒(c) applies and so M is an absolute subgenerator in σ[M].
Every projective module P ∈ σ[M] is isomorphic to a submodule of some M(Λ) which is an absolute subgenerator (by (c)). HenceP is also an absolute subgenerator.
2
Applying 13.5 to M =A, we immediately have the following 13.6 Characterization of left strongly prime rings.
For the ring A the following properties are equivalent:
(a) A is a left strongly prime ring;
(b) every left A-module which is not singular is a subgenerator in A-Mod;
(c) every non-singular left A-module is an absolute subgenerator in A-Mod;
(d) for every non-zero left A-module N,
S(N) = \{K ⊂N | N/K is an absolute subgenerator in A-Mod};
(e) there exists an absolute subgenerator in A-Mod.
The next lemma is needed for the study of strongly prime modules with uniform submodules.
13.7 Lemma. Assume M to be a finitely generated, M-projective A-module and HomA(M, K) 6= 0, for all nonzero K ⊂ M. Then M contains a uniform submodule if and only if S =EndA(M) contains a uniform left ideal.
Proof. For a uniform submodule U ⊂ M consider two nonzero left ideals I, J of S contained in HomA(M, U). ThenU ⊃M I∩M J 6= 0 and
I∩J ⊃HomA(M, M I ∩M J)6= 0, i.e., HomA(M, U) is a uniform left ideal ofS.
By a similar argument one finds that for a uniform left ideal I of S, M I is a
uniform submodule of M. 2
13.8 Strongly prime modules with uniform submodules.
For a finitely generated, self-projective and strongly prime A-module M, the fol-lowing assertions are equivalent:
(a) AM contains a uniform submodule;
(b) AM has finite uniform dimension;
(c) AMc is a finite direct sum of isomorphic indecomposable modules;
(d) S=EndA(M) contains a uniform left ideal;
(e) T =EndA(Mc) is a left artinian simple ring.
Proof. (a) ⇔ (b) For a uniform submodule U ⊂M, we have M ⊂ Uk, k ∈ IN, and hence M has finite dimension.
(a)⇔(c) From the above argument we obtain Mc ⊂Ubk and the assertion follows (since EndA(Ub) is a local ring).
(a)⇔(d) was shown in Lemma 13.7.
(c)⇒(e) Assume Mc = Vk, k ∈ IN, where V is M-injective and uniform. Any submodule ofV is essential, hence rational, sinceMccontains no singular submodules.
ThereforeEndA(V) =: Dis a division ring.
Now T =EndA(Vk) = D(k,k) is a matrix ring over D.
(e)⇒(b) is evident. 2
13.9 Remarks. Primeness conditions on modules.
Consider the following properties of an A-module M.
(i) A/AnA(M) is cogenerated by every 06=K ⊂M; (ii) for every 06=K ⊂M, M ∈σ[K];
(iii) for every 06=K ⊂M, there is a monomorphism M →Kr, r∈IN;
(iv) for every 06=K ⊂M, there is a monomorphism A/AnA(M)→Kr, r∈IN; (v) M is cogenerated by every 06=K ⊂M;
(vi) M is monoform.
For any M, (iii)⇒(v)⇒(i), (iii)⇒(ii)⇒(i) and (iv)⇒(ii).
If M is finitely generated and M-projective, then (ii)⇒(iii).
If M is finitely cogenerated, (ii), (iii) and (v) are equivalent to M being a finite direct sum of isomorphic simple modules.
A direct sum of isomorphic simple modules satisfies (i), (ii), (iii) and (v) but not necessarily (iv) or (vi).
For M = AA, (ii), (iii) and (iv) are equivalent and characterize the left strongly prime rings or ATF-rings as studied by Viola-Prioli [261], Rubin [237], Handelman-Lawrence [150].
For a commutative ring A all these conditions for AA are equivalent to A being a prime ring.
Restricting the above conditions to essential submodules one gets generalizations of commutative semiprime rings. This will be considered in the next section.
Modules with (i) are theprime modulesalready studied in Johnson [168]. Modules with (ii) are called strongly prime modules in Beachy [64]. Rings having a faithful module of this type are the endoprimitive rings of Desale-Nicholson [118].
Modules with (iii) are calledsemicompressible modulesin Beachy-Blair [67]. They are special cases of (ii). Part (1) of 13.4 implies [67, Theorem 3.5]. For M = A, (a)⇔(e) in 13.8 yields [67, Theorem 3.3].
As pointed out in Handelman-Lawrence [150], a ring A is left strongly prime if and only if there is a faithfulA-module satisfying (iv) (SP-modules in [150]). This is equivalent to the existence of a cofaithful strongly prime A-module (i.e., with (ii)).
Condition (v) is the defining property of prime modules in Bican [79]. They are also characterized by the fact that LHomA(M, K) 6= 0 for all nonzero submodules K, L of M.
Modules with condition (vi) (see 11.3) are calledstrongly uniformin Storrer [249].
IfM is monoform and HomA(M, K)6= 0 for every 0 6=K ⊂M, then
K ·EndA(Mc)⊃M ·HomA(M, K)·EndA(Mc) =M ·EndA(Mc) =M ,c
and hence M satisfies (ii) (see also 13.3).
For an extensive account and examples for left strongly prime rings we refer to the monograph [16] by Goodearl, Handelman and Lawrence. Notice that a left strongly prime ring need not be right strongly prime (see [150, Example 1]).
13.10 Exercises.
Recall that A denotes an associative algebra with unit.
(1) Let M be a strongly prime A-module with non-zero socle.
Prove thatM 'E(Λ), whereE is a simple A-module.
(2) LetM be a uniform and self-injective A-module. Prove that the following are equivalent ([85, Proposition 2.1]):
(a) M is strongly prime;
(b) every cyclic submodule of M is strongly prime;
(c) M has no non-trivial self-injective submodules.
(3) Let A be a left self-injective ring. Prove that A is left strongly prime if and only ifA is a simple ring.
(4) Prove that for a ringA, the following are equivalent ([261, 2.1]):
(a) A is left strongly prime;
(b) every finitely generated projective module is a subgenerator in A-Mod;
(c) every non-zero left ideal generates the injective hull of AA.
(5) Let A be a left strongly prime ring. Show ([261]):
(i)A is a left non-singular prime ring.
(ii) A is left noetherian if and only if it contains a non-zero noetherian left ideal.
(iii) Any matrix ring A(n,n), n ≥1, is left strongly prime.
(iv) The centre of A is a (strongly) prime ring.
(v) Every self-injective non-singular left A-module is A-injective ([67]).
(6) Let A be a ring such that AA has finite uniform dimension. Prove that the following are equivalent:
(a) A is left strongly prime;
(b) A is a left non-singular prime ring.
(7) Prove that for a left strongly prime ringA, the following are equivalent ([261]):
(a) A is a simple ring;
(b) every left ideal in A is idempotent.
(8) Prove: A regular ring is simple if and only if it is left (right) strongly prime.
(9) Prove that for a ringA, the following are equivalent ([261]):
(a) A is a simple left artinian ring;
(b) A is left strongly prime and SocAA 6= 0.
(10) Let Abe a ring which is finitely generated as a module over its centre. Prove that the following are equivalent ([237, 1.10]):
(a) A is left strongly prime;
(b) A is a prime ring;
(c) A is right strongly prime.
(11) Prove that for any ring A, the following are equivalent ([230, Proposition 1]):
(a) Every self-injective left A-module isA-injective (A is left QI);
(b) A is left noetherian and any non-zero uniform (injective) A-module is strongly prime.
(12) Prove that for any ring A, the following are equivalent:
(a) A has a faithful strongly prime module;
(b) A has a faithful (self-injective) left module with no non-trivial fully invariant submodules;
(c) A has a left ideal L⊂A, such thatA/L is faithful and satisfies:
For anya ∈A,a0 ∈A\L, there existr1, . . . , rk∈A, such that rria ∈L, for r∈A and alli≤k, implies ra0 ∈L.
Rings with these properties are called left endoprimitive ([118, 2.1]).
(13) A ring is calledleft completely torsion free (CTF)if, for any hereditary torsion classes T inA-Mod, T(A) = 0 orA .
Prove that the following are equivalent ([16, 13.1]):
(a) A is left CTF;
(b) every non-zero injective left A-module is faithful;
(c) for each non-zero two-sided ideal K ⊂A, there exists k ∈K such that HomA(Ak, A/K)6= 0.
(14) Prove that the following are equivalent ([176, 2.4]):
(a) A is left CTF and Soc(AA)6= 0;
(b) every non-zero injective left A-module is a cogenerator;
(c) there are only two hereditary torsion classes inA-Mod;
(d) A is isomorphic to some S(n,n), n ∈IN, whereS is a local left perfect ring.
(15) Prove for the algebra A ([176]):
(i) The following assertions are equivalent:
(a) Every non-zero left ideal ofA is a generator in A-Mod;
(b) every non-zero ideal of A is a generator inA-Mod;
(c) every non-zero submodule of a projective A-module generates A.
Algebras with this properties are called left G-algebras.
(ii) If SocAA 6= 0 then A is a left G-algebra if and only if it is left artinian and simple.
(iii) Every left G-algebra is left strongly prime. Any left self-injective and left strongly prime algebra is a left G-algebra.
(16) A leftA-moduleM is calledcoprimeifM is generated by each of its non-zero factor modules. Prove ([79]):
(i)M is coprime if and only if it is generated by every non-zero cocyclic factor module.
(ii) Every direct sum of copies of a simple module is a coprime module.
(iii) Any coprime module M with RadM 6=M is semisimple.
(iv) EveryA-module is coprime if and only ifA is isomorphic to a matrix ring over a division ring.
References. Beachy [64], Beidar-Wisbauer [75, 76, 77], Bican-Jambor-Kepka-Nemec [79], Handelman-Lawrence [150], Katayama [176], Rubin [237], Viola-Prioli [261], Wisbauer [274].