Modules Close to SSP- and SIP-Modules
Adel N. Abyzov*, Tran Hoai Ngoc Nhan**, and Truong Cong Quynh***
(Submitted by M. M. Arslanov)
Department of Algebra and Mathematical Logic, Kazan (Volga Region) Federal Univesrity, 18 Kremlyovskaya str., Kazan, 420008 Russia,
Department of IT and Mathematics Teacher Training, Dong Thap University, Vietnam, Department of Mathematics, Danang University, 459 Ton Duc Thang, Danang city, Vietnam
Received June 22, 2016
Abstract—In this paper, we investigate some properties of SIP, SSP and CS-Rickart modules. We give equivalent conditions for SIP and SSP modules; establish connections between the class of semisimple artinian rings and the class of SIP rings. It shows thatR is a semisimple artinian ring if and only ifRRis SIP and every rightR-module has a SIP-cover. We also prove thatRis
a semiregular ring andJ(R) =Z(RR)if only if everyfinitely generated projective module is a
CS-Rickart module which is also aC2module. DOI:10.1134/S1995080217010024
Keywords and phrases: SSP-module, SIP-module, SIP-CS, CS-Rickarts.
1. INTRODUCTION AND NOTATION
Throughout this paperRdenotes an associative ring with identity, and modules will be unitary right
R-modules. The Jacobson radical ideal inR is denoted byJ(R). The notations N ≤M, N ≤eM,
N M, orN ⊂dMmean thatN is a submodule, an essential submodule, a fully invariant submodule, and a direct summand ofM, respectively. We refer to [6, 9, 18], and [23] for all the undefined notions in this paper.
Recall that a moduleMis called a SIP module (respectively, SSP module) if the intersection (or the sum) of any two direct summands ofMis also a direct summand ofM(see [12, 14, 22]). It is known that every Rickart rightR-moduleM(i.e., every endomorphism ofM has the kernel a direct summand) has the SIP (see [16, Proposition 2.16]) and every d-Rickart rightR-moduleM (i.e., every endomorphism ofMhas the image a direct summand) has the SSP ([17, Proposition 2.11]).
A moduleM is called an SIP-CS module if the intersection of any two direct summands of M is essential in a direct summand ofM. It is known that every CS-Rickart module has the CS-SIP (see [2, Proposition 1.(4)]).
In this paper, we provide some characterizations of SIP, SSP, SIP-CS and CS-Rickart modules.
*E-mail:[email protected]
**E-mail:[email protected] ***E-mail:[email protected]
2. SIP MODULES AND SSP MODULES
Letf :A→B be a homomorphism. We denote byf the submodule ofA⊕B as follows: f= {a+f(a)|a∈A}. The following result is obvious and we can omit its proof.
Lemma 2.1. LetM =X⊕Y andf :A→Y, a homomorphism withA≤X. Then (1)A⊕Y =f ⊕Y;
(2) Ker(f) =X∩ f.
We next study some properties of SIP and SSP modules via homomorphisms:
Proposition 2.2. The following conditions are equivalent for a moduleM: (1)M is SSP;
(2) For any split monomorphismf :A→M withAa direct summand ofM,A+Im(f)is a direct summand ofM;
(3) For any split epimorphismf :M →M/AwithAa direct summand ofM,A+Ker(f)is a direct summand ofM.
Proof.(1)⇒(2),(3)are obvious.
(2)⇒(1). Assume that M =A1⊕A2 and f :A1 →A2 an R-homomorphism. Call T =f a
submodule of M and hence M =T⊕A2. We consider the homomorphism ψ:A1 →M given by
ψ(x) =x+f(x). It is easily to see thatψis a split monomorphism. By (2),A1+ψ(A1) =A1+Tis a
direct summand ofM. Furthermore,A1+T =A1⊕Im(f), which implies Im(f)is a direct summand
ofA2.
(3)⇒(1). Suppose thatM =A1⊕A2 andf :A1 →A2 anR-homomorphism. LetT =fbe a
submodule ofM. ThenM =T⊕A2. Call the homomorphismψ:M →M/T given byψ(a1+a2) =
a2+T for alla1 ∈A1, a2 ∈A2. Clearly, ψis a split epimorphism and Ker(ψ) =A1. By (3), A1+T
is a direct summand ofM. On the other hand, A1+T =A1⊕Im(f), which implies Im(f)is a direct
summand ofA2. 2
Corollary 2.3.The following conditions are equivalent for a moduleM: (1)M is SSP;
(2) For any two direct summandsA1andA2withA1 A2, thenA1+A2is a direct summand ofM.
Similarly with SIP, we also have some characterizations of SIP-modules:
Proposition 2.4. The following conditions are equivalent for a moduleM: 1. MisSIP;
2. For any split monomorphism f :A→M with A a direct summand of M, A∩f(A) is a direct summand ofM;
3. For any split epimorphismf :M → M/AwithAa direct summand of M, A∩Ker(f)is a direct summand ofM.
Corollary 2.5.The following conditions are equivalent for a moduleM: 1. Mis SIP;
Proposition 2.6. Let R be a ring, M anR-R-bimodule and T =R∝M the corresponding trivial extension. The following conditions are equivalent: (1)Thas the SSP;
(2) (a) R has the SSP;
(b) For every regularxofRwithx=xyx, we havexM(1−xy) = 0. Proof.(1)⇒(2). By [12, Proposition 4.5].
(2)⇒(1). Assume that x=xyx. For any m∈M, call z=xm(1−xy). It follows that z= (xy)z(1−xy). Note thatxyis idempotent ofR. By [12, Proposition 4.5],z= (xy)z(1−xy) = 0. 2 Let R be a ring and Ω, a class of right R-modules which is closed under isomorphisms and summands. According to Enochs in [10], we study the notion ofΩ-envelope and the notion ofΩ-cover: AnR-homomorphismg:M →Eis called anΩ-envelope of a rightR-moduleM; ifE ∈Ωsuch that any diagram:
withE ∈Ω, can be completed, and the diagram:
can be completed only by an automorphismh.
AnR-homomorphismg:E →Mis called anΩ-cover of a right R-moduleM; ifE ∈Ωsuch that any diagram:
withE ∈Ω, can be completed, and the diagram:
can be completed only by an automorphismh.
A rightR-moduleM is called aC3-moduleif wheneverA andB are direct summands ofM with
A∩B= 0, thenA⊕Bis a direct summand ofM. Dually,M is called aD3-moduleif wheneverM1
andM2are direct summands ofMand M =M1+M2,thenM1∩M2is a direct summand ofM.
Proposition 2.7. The following conditions are equivalent for a ringR: (1)Ris a semisimple artinian ring;
(2) Every rightR-module has a D3-cover;
(3) Every2-generated rightR-module has aD3-cover; (4) Every rightR-module has aD3-envelope;
(5) Every2-generated rightR-module has aD3-envelope. Proof.(1)⇒(2)⇒(3). Clear.
(3)⇒(1). Let S be a simple right R-module. Call ϕ:RR→S an epimorphism. By (3), M = RR⊕Shas a D3-cover, sayα:C→MwhereCis a D3-module. Letι1:S →Mandι2 :RR→Mbe
the inclusion maps for alli= 1,2. Note thatSandRRare D3-modules, and there are homomorphisms β1 :S →C, β2 :RR→Csuch thatαβi =ιi. Clearly,idM =ι1⊕ι2=α(β1⊕β2). This shows thatM
is isomorphic to a direct summand ofC, which implies thatM is a D3-module. We deduce that Ker(ϕ) is a direct summand ofRRby [4, Proposition 4]. It follows that S is a projective module. ThusR is semisimple.
(1)⇒(4)⇒(5). Clear.
(5)⇒(1). Let S be a simple right R-module. Call ϕ:RR→S an epimorphism. By (5), M = RR⊕S has a D3-envelope, named ι:M →E where E is a D3-module. Since S and R are
D3-modules, there existf1 :E →S, f2:E →Rsuch thatfiι=πi, whereπ1:M →S andπ2:M →R
are the projections. There existsφ:E→Msuch thatπiφ=fifor alli= 1,2. It follows thatφι=idM,
and henceιis a split monomorphism. ThusN⊕E(N)is isomorphic to a direct summand ofE. This gives thatS⊕Ris also a D3-module. We deduce that Ker(ϕ) is a direct summand ofRR. So S is a
projective module. ThusRis semisimple. 2
Corollary 2.8.The following conditions are equivalent for a ringR: (1)Ris a semisimple artinian ring;
(2)RRis SIP and every rightR-module has a SIP-cover;
(3)RRis SIP and every2-generated rightR-module has a SIP-cover;
(4)RRis SIP and every rightR-module has a SIP-envelope;
(5)RRis SIP and every2-generated rightR-module has a SIP-envelope.
A ringRis called a right V-ring if every simple rightR-module is injective.
Proposition 2.9. The following conditions are equivalent for a ringR: (1)Ris a rightV-ring;
(2) Everyfinitely cogenerated rightR-module has aC3-envelope; (3) Everyfinitely cogenerated rightR-module has aC3-cover. Proof.(1)⇒(2),(3)are obvious.
(2)⇒(1)LetN be an arbitrary simple module. Assume thatι:M =N ⊕E(N)→E is the C3-envelope, whereE is a C3-module. SinceN andE(N) are C3-modules, there existf1:E →N, f2:
E →E(N) such that fiι=πi, where π1 :M →Ni and π2 :M →E(N) are the projections. There
existsφ:E→Msuch thatπiφ=fifor alli= 1,2. It follows thatφι=idM, and so the monomorphism ιsplits. ThusN⊕E(N)is isomorphic to a direct summand ofE. It follows thatN ⊕E(N)is also a C3-module. ThereforeN is a direct summand ofE(N). This givesN is injective. ThusRis a right V-ring.
(3)⇒(1)The proof is similar to the proof(3)⇒(1)of Proposition 2.7. 2 Similarly, we also get the following result for injectivity of semisimple modules:
Proposition 2.10. The following conditions are equivalent for a ringR: (1)Ris a right Noetherian rightV-ring;
(2) Every rightR-module with essential socle has aC3-envelope; (3) Every rightR-module with essential socle has a C3-cover.
3. SIP-CS MODULES
A moduleMis called relatively CS-Rickart toN (orN-CS-Rickart) if for everyϕ∈EndR(M, N), Kerϕis an essential submodule of a direct summand ofM. A moduleM is called relatively d-CS-Rickart toN (orN-d-CS-Rickart) if for everyϕ∈EndR(N, M), Imϕlies above a direct summand
ofM. A moduleM is called CS-Rickart (d-CS-Rickart) ifM isM-CS-Rickart (resp., M -d-CS-Rickart). Mis called a SIP-CS module ifAiis essential in a direct summand ofM for alli∈ I,Iis a
finite index set, theni∈IAiis essential in a direct summand ofM.Mis called a lifting SSP module if
Ailies above a direct summand ofMfor alli∈ I,Iis afinite index set, then
i∈IAilies above a direct
summand ofM. The class of CS-Rickart (d-CS-Rickart, SIP-CS, lifting SSP) modules is studied by the authors in [1, 2].
Lemma 3.1. The following implications hold for a moduleM =M1⊕. . .⊕Mn:
(1) ifM is relativelyCS-Rickart toN thenMirelativelyCS-Rickart toN; (2) ifM is relativelyd-CS-Rickart toNthenMirelativelyd-CS-Rickart toN. Proof.We only need to prove for the casen= 2,i= 1.
(1) Assume thatM =M1⊕M2 is relatively CS-Rickart toN. There existsϕ:M →N such that
ϕ=ψ⊕0|M2 for eachψ:M1→N. By assumption, there exists a direct summandDofM such that
Ker(ϕ)≤e D. SinceKer(ϕ) = Ker(ψ)⊕M2andD= (D∩M1)⊕M2, it follows thatKer(ψ)⊕M2≤e
(D∩M1)⊕M2. ThereforeKer(ψ)≤eD∩M1. SinceDis a direct summand ofM,D∩M1is a direct
summand ofM1. HenceM1is relatively CS-Rickart toN.
(2) Assume thatM =M1⊕M2is relatively d-CS-Rickart toN. There existsϕ:M →Nsuch that
ϕ=ψ⊕0|M2 for eachψ:M1→N. By assumption,Imϕ= Imψlies above a direct summand ofN.
Thus,M1is relatively d-CS-Rickart toN. 2
Proposition 3.2. The following implications hold for a moduleM:
(1) if M is a SIP-CS module with C2condition andM =M1⊕M2 then M1 relatively
CS-Rickart toM2;
(2) if M is a liftingSSP module withD2 condition andM =M1⊕M2 thenM1 relativelyd
-CS-Rickart toM2.
Proof.Letf :M1 →M2be anR-homomorphism. ThenM =f ⊕M2.
(1)We have that Ker(f) =f ∩M1 ≤eeM for somee2 =e∈S, byM is SIP-CS. Letπ1:M1⊕
M2 →M1be the canonical projection and henceeM ∩M2 = 0, implies thatπ1(eM)∼=eM. SinceMis
aC2module,π1(eM)is a direct summand ofM. Then, since Ker(f)≤eeM, Ker(f) =π1(Ker(f))≤e
π1(eM). Hence,M1is relatively CS-Rickart toM2.
(2) We have that Im(f)⊕M1=f+M1 lies above eM for some e2 =e∈S, by M is lifting
SSP. Since f+M1+M2 =M, eM +M2 =M by [8, 3.2.(1)]. Let π1:M1⊕M2 →M1 be the
canonical projection. Then π1|eM splits byD2 condition. It follows that eM = (eM ∩M2)⊕N by
Ker(π1|eM) =eM ∩M2. We haveN⊕M2 =N+ (eM ∩M2+M2) =eM +M2 =Mand obtain that
N ⊕Im(f) =M1⊕Im(f)⊃eM. By modular law, eM =N⊕eM ∩Im(f). As Im(f)⊕M1 eM M eM, we have that N⊕Im(f) N ⊕eM ∩Im(f) N ⊕M2 N ⊕eM ∩Im(f). This is equivalent to Im(f) eM ∩Im(f) M2 eM∩Im(f), which implies that Im(f)lies above the direct summandeM ∩Im(f)ofM. 2
Corollary 3.3.The following implications hold for a moduleM =M1⊕. . .⊕Mn:
(1) ifM is a SIP-CS module with C2condition thenMi is relatively CS-Rickart toMj for every i=j;
(2) ifM is a lifting SSP withD2condition thenMiis relatively d-CS-Rickart toMj for everyi=j. Proof. IfM is a SIP-CS module withC2condition (respectively, lifting SSP withD2condition), then by Proposition 3.2,i=jMiis relatively CS-Rickart toMj(respectively, relatively d-CS-Rickart toMj). By Lemma 3.1,Miis relatively CS-Rickart toMj (respectively, relatively d-CS-Rickart toMj)
for everyi=j. 2
(1) ifM ⊕M is a SIP-CS module withC2condition thenM is a CS-Rickart module; (2) ifM ⊕M is a lifting SSP withD2condition thenMis a d-CS-Rickart module.
Proof.Follow from Corollary 3.3. 2
The singular submoduleZ(M) of a rightR-moduleM is defined asZ(M) ={m∈M :annrR(m) is an essential right ideal ofR}whereannrR(m)denotes the right annihilator ofminR. The singular submodule ofRR is called the (right) singular ideal of the ringR and is denoted byZ(RR). It is well known thatZ(RR)is indeed an ideal ofR.
Next we give a necessary and sufficient condition for a ring over which every finitely generated projective module to be a SIP-CS-module which is also aC2module.
Theorem 3.5. The following conditions are equivalent for a ringR: (1)Ris a semiregular ring andJ(R) =Z(RR);
(2) Every finitely generated projective module is a CS-Rickart module which is also a C2 module;
(3) Everyfinitely generated projective module is aSIP-CS module which is also aC2module; (4) Everyfinitely generated projective module is aSIP-CS module which is also aC3module. Proof.(1)⇒(2). Follows from [2, Theorem 2].
(2)⇒(3). Follows from [2, Proposition 1].
(3)⇒(2). Let P be a finitely generated projective module. By the hypothesis, P is a SIP-CS module which is also aC2module. ThenP ⊕P is a SIP-CS module which is also aC2module. Since Proposition 3.2,Pis relatively CS-Rickart toP, it means thatPis a CS-Rickart module.
(3)⇔(2). Follows from [1, Corollary 3.5]. 2
Lemma 3.6. The following conditions are equivalent for a moduleM: (1)M is aSIP-CS module;
(2) Intersection of every pair of direct summands ofM is essential in a direct summand ofM.
Proof.It is obvious. 2
Proposition 3.7. Assume that M is a SIP-CS module. Then for any decomposition M = M1⊕M2andf :M1 →M2is a homomorphism, thenKer(f)is essential in a direct summand of
M.
Proof. Assume that M =M1⊕M2 and f :M1 →M2 an R-homomorphism. Call T =f a
submodule ofM. SoM =T⊕M2 and Ker(f) =T∩M1. On the other hand, by the hypothesis,M
is a SIP-CS and hence Ker(f)is essential in a direct summand ofMby Lemma 3.6. 2
Corollary 3.8. LetM be a module andN, a nonsingular module. IfM⊕N is a SIP-CS module, then every homomorphism fromM toN has the kernel a direct summand ofM.
Proof. Letf :M →N be a non-zero homomorphism. By Proposition 3.7, Ker(f) is essential in a direct summand ofM ⊕N. Assume thatAis a direct summand ofM⊕N such that Ker(f)≤eA. Call πM :M⊕N →M the canonical projection and h= (f ◦πM)|A:A→N. Therefore Ker(h) =
Ker(f)⊕(N∩A). We have that Ker(f)≤eAand obtain that Ker(f)⊕(N ∩A)≤eA. It follows that Ker(f)⊕(N∩A) =A. Thus Ker(f)is a direct summand ofM. 2
Corollary 3.9.LetMbe an indecomposable module andNbe a nonsingular module. IfM ⊕N is a SIP-CS module, then every nonzero homomorphism fromM toN is a monomorphism.
Proposition 3.10. LetMbe a nonsingular rightR-module. If(R⊕M)Ris aSIP-CS module, then every cyclic submodule ofM is projective.
Proof. Letmbe a non-zero arbitrary element ofM. Call the homomorphismϕ:RR→M given by
ϕ(x) =mx. As(R⊕M)Ris a SIP-CS module, Ker(ϕ)is a direct summand ofRRby Corollary 3.8. It
follows that Im(ϕ)is isomorphic to a direct summand ofRR. ThusmRis a projective module. 2 A ringRis called right (semi)hereditary if every (finitely generated) right ideal ofRis projective.
Theorem 3.11. The following statements are equivalent for a right nonsingular ringR: (1)Ris right hereditary;
(2) Every projective rightR-module is aSIP-CS module. Proof.(1)⇒(2)is obvious.
(2)⇒(1)LetIbe a right ideal ofR. We will show thatIis a projective module. Call an epimorphism
ϕ:F →N for some free rightR-moduleF. Letιbe the inclusion map from I toRR. Consider the homomorphismι◦ϕ:F →RR. By (2), F⊕RR is a SIP-CS module. We have from Corollary 3.8,
Ker(ϕ) =Ker(ι◦ϕ)is a direct summandF. This gives thatF =Ker(ϕ)⊕Bfor some submoduleBof
F. Thus,I is projective. 2
The author Warfied proved that ifRis right serial, thenRis right nonsingular if and only ifRis right semihereditary.
The same argument of the proof of Theorem 3.11, we also have the following result of semihereditary rings:
Theorem 3.12. The following statements are equivalent for a right nonsingular ringR: (1)R is right semihereditary;
(2) Everyfinitely generated projective rightR-module is aSIP-CS module; (3) Everyfinitely generated free rightR-module is aSIP-CS module. LetM be a rightR-module andS =End(M). We denote
Δ(S) ={f ∈S|Ker(f)≤eM}.
AnR-module is called a self-generator if it generates all its submodules.
Theorem 3.13. The following conditions are equivalent for a self-generator moduleM with S =End(M):
(1)Sis a semiregular ring withJ(S) = Δ(S); (2)M is aCS-Rickart and C2 module.
Proof. (1)⇒(2)Assume thatSis a semiregular ring withJ(S) = Δ(S). AsMis a self-generator,
J(S) = Δ(S)≤Z(SS). We deduce that S is right C2. This gives that M is a C2-module by [20,
Theorem 7.14(1)]. Let α:M →M be an endomorphism of M. As S is a semiregular ring, there existsβ ∈Ssuch thatβ =βαβandα−αβα∈J(S)by [19, Theorem 2.9]Ni. Calle= 1−βα. Then
e2 =e∈S. Asα−αβα∈Δ(S), Ker(α−αβα)≤eM and hence Ker(α−αβα)∩e(P)≤ee(M). It is easily to check that Ker(α−αβα)∩e(M) =Ker(α). We deduce that Ker(α)≤ee(M).
(2)⇒(1)By [21, Theorem 3.2]. 2
Corollary 3.14. The following conditions are equivalent for a self-generator moduleM with S= End(M(N)):
(1)Sis a semiregular ring withJ(S) = Δ(S); (2)M(N)is a CS-Rickart and C2 module; (3)M(N)is a SIP-CS and C2 module.
Proof.By Proposition 3.2 and Theorem 3.13. 2
ACKNOWLEDGMENTS
The third author was supported by the Russian Government Program of Competitive Growth of Kazan Federal University. The author would like to thank the members of Department of Algebra and Mathematical Logic, Kazan (Volga Region) Federal University for their hospitality.
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