PII. S0161171204302073 http://ijmms.hindawi.com © Hindawi Publishing Corp.
FLAT SEMIMODULES
HUDA MOHAMMED J. AL-THANI
Received 8 February 2003
To my dearest friend Najla Ali
We introduce and investigate flat semimodules andk-flat semimodules. We hope these con-cepts will have the same importance in semimodule theory as in the theory of rings and modules.
2000 Mathematics Subject Classification: 16Y60.
1. Introduction. We introduce the notion of flat and k-flat. InSection 2, we study the structure ensuing from these notions.Proposition 2.4asserts thatV is flat if and only if (V⊗R−) preserves the exactness of all right-regular short exact sequences.
Proposition 2.5gives necessary and sufficient conditions for a projective semimodule to be k-flat. In Section 3, Proposition 3.3 gives the relation between flatness and in-jectivity. InSection 4,Proposition 4.1characterizes thek-flat cancellable semimodules with the left ideals.Proposition 4.4describes the relationship between the notions of projectivity and flatness for a certain restricted class of semirings and semimodules. Throughout,Rwill denote a semiring with identity 1. All semimodulesMwill be left R-semimodules, except at cited places, and in all cases are unitary semimodules, that is, 1·m=mfor allm∈M (m·1=mfor allm∈M)for all leftR-semimodulesRM (resp., for all rightR-semimoduleMR).
We recall here (cf. [1,2,4,7,8]) the following facts.
(a) A semiringRis said to satisfy the left cancellation law if and only if for alla,b,c∈ R,a+b=a+c⇒b=c. A semimoduleMis said to satisfy the left cancellation law if for allm,m,m∈M,m+m=m+m⇒m=m.
(b) We say that a nonempty subsetN of a left semimoduleM is subtractive if and only if for allm,m∈M,m,m+minNimplyminN.
(c) A semiring R is called completely subtractive ifRR is a completely subtractive
semimodule; and a leftR-semimoduleMis called completely subtractive if and only if for every subsemimoduleNofM,Nis subtractive.
(d) A semimoduleMis said to be freeR-semimodule ifMhas a basis overR. (e) A semimoduleC is said to be semicogenerated byUwhen there is a homomor-phismϕ:M→ΠACsuch that kerθ=0. A semimoduleCis said to be a semicogenerator
whenCsemicogenerates every leftR-semimoduleM.
said to be a semimonomorphism if kerα=0, to be a semi-isomorphism ifαis surjective and Kerα=0, to be an isomorphism ifαis injective and surjective, to bei-regular if α(M)=Imα, to bek-regular if fora,a∈A,α(a)=α(a)implyinga+k=a+kfor
somek,k∈kerα, and to be regular if it is bothi-regular andk-regular.
(g) AnR-semimoduleMis said to bek-regular if there exist a freeR-semimoduleF and a surjectiveR-homomorphismα:F→Msuch thatαisk-regular.
(h) The sequence K α→M β→N is called an exact sequence if Kerβ=Imα, and proper exact if Kerβ=α(K).
(i) A short sequence 0→K α→M β→N→0 is said to be leftk-regular right regular ifαisk-regular andβis right regular.
(j) For any two R-semimodules N, M, HomR(N,M) := {α : N → M | αis an R-homomorphism of semimodules}is a semigroup under addition. IfM,N, andUare R-semimodules andα:M→Nis a homomorphism, then Hom(α,IU): HomR(N,U)→
HomR(M,U)is given by Hom(α,IU)γ=γα, whereIU is the identity onU.
(k) If M is a right R-semimodule, N is a left R-semimodule, and T is an N -semi-module, then a functionθ:M×N→T isR-balanced if and only if, for allm,m∈M,
for alln,n∈N, and for allr∈R,we have (1) θ(m+m,n)=θ(m,n)+θ(m,n),
(2) θ(m,n+n)=θ(m,n)+θ(m,n),
(3) θ(mr ,n)=θ(m,r n).
LetRbe a semiring, letMbe a rightR-semimodule, and letNbe a leftR-semimodule. LetAbe the setM×N, and letUbe theN-semimodule⊕AN×⊕AN. LetWbe the subset
ofUconsisting of all elements of the following forms: (1) (α[m+m,n],α[m,n]+α[m,n]),
(2) (α[m,n]+α[m,n],α[m+m,n]),
(3) (α[m,n+n],α[m,n]+α[m,n]),
(4) (α[m,n]+α[m,n],α[m,n+n]),
(5) (α[mr ,n],α[m,r n]), (6) (α[m,r n],α[mr ,n]),
form and m inM, nand n in N, and r inR, and where α[m,n] is the function
fromM×NtoNwhich sends(m,n)to 1 and sends every other element ofM×Nto 0. LetUbe theN-subsemimodule ofUgenerated byW. DefineNcongruence relation ≡on⊕ANby settingα≡αif and only if there exists an element(β,γ)∈Usuch that
α+β=α+γ. The factorN-semimodule⊕AN/≡will be denoted byM⊗RN, and is
called the tensor product ofMandNoverR.
(A) A leftR-semimodulePis said to be projective semimodule if and only if for each surjectiveR-homomorphismϕ:M→N, the induced homomorphismϕ: HomR(P,M)→
HomR(P,N)is surjective.
2. Flat andk-flat semimodules. In this section, we discuss the structure of flat andk -flat semimodules.Proposition 2.4asserts thatVis flat if and only if(V⊗R−)preserves
Definition2.1. A semimoduleVRis flat relative to a semimoduleRM(or thatV is
M-flat) if and only if for every subsemimoduleK≤M, the sequence 0→V⊗RK IV⊗iK→
V⊗RM is proper exact (i.e., Ker(IV⊗RiK)=0) whereIV⊗RiK(v⊗k)=v⊗iK(k). A
semimodule VR that is flat relative to every leftR-semimodule is called a flat right
R-semimodule.
Definition2.2. A semimoduleVRisk-flat relative to a semimoduleRM(or thatVis
Mk-flat) if and only if for every subsemimoduleK≤M, the sequence 0→V⊗RK IV⊗RiK→
V⊗RMis proper exact andIV⊗iKisk-regular (i.e.,IV⊗RiKis injective). A semimoduleVR
that isk-flat relative to every rightR-semimodule is called ak-flat rightR-semimodule. Thus, ifVRisk-flat relative toRM, thenVRis flat relative toRM.
Our next result shows that the class of flat andk-flat semimodules is closed under direct sums.
Proposition2.3. Let(Vα)α∈Abe an indexed set of rightR-semimodules. Then⊕AVα
isM-flat (k-flat) if and only if eachVαisM-flat (k-flat).
Proof. LetM be a leftR-semimodule andKa subsemimodule ofM. Consider the
following commutative diagram:
Vα⊗K
IV α⊗iK
iα
Vα⊗M
πα
⊕Vα⊗K θ
πα
ϕ
⊕Vα⊗M
ϕ
iα
⊕Vα⊗K
I⊕V α⊗iK
⊕Vα⊗M,
(2.1)
whereπα:⊕(Vα⊗K)→Vα⊗K andiα:Vα⊗K→ ⊕(Vα⊗K)are defined respectively
byπα :(vα⊗kα)vα⊗kα and iα :vα⊗kα(vi⊗ki), wherevi⊗ki=0 if i≠α and
vi⊗ki=vα⊗kα ifα=i;ϕandϕare the isomorphisms of [8, Proposition 5.4] given
byϕ[(vα)⊗k]=(vα⊗k)and θ(vα⊗k)=(vα⊗i(k)). Now suppose that⊕Vα is M
-flat (k-flat). IfIVα⊗iK(vα⊗k)=0[IVα⊗iK((vα⊗k))=IVα⊗iK((vα⊗k))], then by the
above diagram we have(vα)⊗iK(k)=0[(vα)⊗i(k)=(vα)⊗i(k)]. Since⊕Vα is flat
(k-flat), then (vα)⊗k=0[(vα)⊗k=(vα)⊗k]. Again by (2.1), (vα⊗k)=0 whence
vα⊗k=0[(vα⊗k)=(vα⊗k),whencevα⊗k=vα⊗k]. ThereforeVαis flat (k-flat).
Conversely, suppose that Vα is M-flat (k-flat) for each α∈A. If I⊕Vα⊗iK((vα)⊗ k)=0[I⊕Vα⊗iK((vα)⊗k)=I⊕Vα⊗iK((vα)⊗k)], then by the above diagram we have
vα⊗i(k)=0[vα⊗i(k)=vα⊗i(k)]for eachα∈A. SinceVαis flat (k-flat), thenvα⊗k=
0[vα⊗k=vα⊗k]for eachα. Therefore,(vα⊗k)=0[(vα⊗k)=(vα⊗k)]. Again by
(2.1),(vα)⊗k=0[(vα)⊗k=(vα)⊗k]. Thus⊕Vαis flat (k-flat).
Proposition2.4. LetMbe a leftR-semimodule. A rightR-semimoduleVisMflat if
short exact sequences with middle termM:
0 →RK α→RM β→RN →0. (2.2)
Proof. “If” part. Let 0→RK α→RM β→RN→0 be a leftk-regular right regular
exact sequence. SinceVRisRM-flat, then using [8, Theorem 5.5(2)], the sequence
0→V⊗RK IV⊗α→V⊗RM IV⊗β→V⊗RN →0 (2.3)
is exact.
“Only if” part. LetRK≤RM. Consider the following exact sequence:
0→K iK→M πImiK→M/Imi
K →0. (2.4)
By hypothesis, 0→V⊗RK IV⊗iK→V⊗RMis an exact sequence. ThusV isM-flat.
Our next result gives a necessary and sufficient condition for a projective semimodule to bek-flat relative to a cancellable semimoduleM.
Proposition2.5. LetVRbe projective andRMcancellable. Then,VisMk-flat if and
only if the functor(V⊗R−)preserves the exactness of all left k-regular right regular
short exact sequences
0 →RK α→RM β→RN →0. (2.5)
Proof. “If” part. Let 0→RK α→RM β→RN→0 be a leftk-regular right regular
exact sequence. SinceVRisRM k-flat, thenVRisRM-flat. By usingProposition 2.4, the
sequence
0→V⊗RK IV⊗α→V⊗RM IV⊗β→V⊗RN →0 (2.6)
is exact.
“Only if” part. LetK≤M. Consider the following exact sequence:
0→K iK→M πImiK→M/Imi
K →0. (2.7)
SinceV is projective andM is cancellable, then by using [9, Proposition 1.16],IV⊗iK
isk-regular. By hypothesis, 0→V⊗RK IV⊗iK→V⊗RMis an exact sequence. ThusV is
Mk-flat.
3. Flatness via injectivity. We will discuss the relation between the injectivity and flatness. By(·)∗we mean the functor Hom
N(−,C), whereCis a fixed injective
semico-generator cancellativeN-semimodule.
Remark3.1. IfUis a rightR-semimodule, thenU∗is a leftR-semimodule.
We state and prove the following lemma, analogous to the one on modules which is needed in the proof ofProposition 3.3.
Lemma 3.2. LetR be a semiring, let M and M be left R-semimodules, and let U
be a rightR-semimodule. LetT be a cancellativeN-semimodule. Ifα:M→M is an
R-homomorphism, then there existN-isomorphismsϕandϕsuch that the following
diagram commutes:
HomRM,HomN(U,T )
ϕ
HomR(α,IHomN(U,T ))
HomRM,HomN
U,TN
ϕ
HomN
U⊗RM,T
HomN((IU⊗α),IT)
HomN
U⊗RM,T.
(3.1)
Proof. By [7, Proposition 14.15], there exists anN-isomorphism
ϕ: HomRM,HomN(U,T )
→HomR(M⊗U,T ) (3.2)
given byϕ(γ):u⊗mγ(m)u. Then with a parallel definition forϕ, we have
ϕHom
Rα,IHomN(U,T )
(γ)(u⊗m) =ϕ(γα)(u⊗m)=(γα)(m)(u) =γα(m)(u)=ϕ(γ)u⊗α(m) =ϕ(γ)IU⊗α(u⊗m)
=HomNIU⊗α,ITϕ(γ)(u⊗m),
(3.3)
and the diagram commutes.
Proposition3.3. LetMbe a leftR-semimodule.
(1) If the rightR-semimoduleV isMk-flat, thenV∗isM-injective.
(2) IfV∗isM-injective, thenVisM-flat.
Proof. (1) LetKbe a subsemimodule ofM. SinceV isMk-flat, then the sequence
0→V⊗K IV⊗iK→V⊗Mis proper exact, andI
V⊗iKisk-regular. ByLemma 3.2, we have
the following commutative diagram:
HomRM,V∗
ϕ
HomR(iK,IV∗)
HomRK,V∗ 0
ϕ
(V⊗M)∗ Hom(IV⊗iK,IC)
(V×K)∗ 0,
(3.4)
whereϕandϕareN-isomorphisms. It follows that the top row is proper exact if and
only if the bottom row is proper exact, whence by [6, Proposition 3.1],V∗is injective. (2) IfV∗is injective, then
is proper exact. Again by the above diagram,
(V⊗M)∗ Hom(IV⊗iK,Ic)
→(V⊗K)∗ →0 (3.6)
is proper exact. Hence, the sequence is exact. SinceCis a semicogenerator, then by [3, Proposition 4.1], the sequence 0→V⊗K→V⊗M is an exact sequence. Hence, V is M-flat.
4. Cancellable semimodules. In this section, we deal with cancellable semimodules. We characterizek-flat cancellable semimodules by means of left ideals.
Proposition4.1. The following statements about a cancellable rightR-semimodule Vare equivalent:
(1) Visk-flat relative toRR;
(2) for each (finitely generated) left ideal I ≤RR, the surjective N-homomorphism
ϕ:V⊗RI→V Iwithϕ(v⊗a)=vais ak-regular semimonomorphism.
Proof. (1)⇒(2). SinceVis cancellable, then by using [7, Proposition 14.16],V⊗RR V. Consider the following commutative diagram:
V⊗RI
ϕ
IV⊗iI
V⊗RR
θ
V I iV I V ,
(4.1)
whereθ is the isomorphism of [7, Proposition 14.16]. Sinceψ:V×I→V Igiven by ψ(v,i)=vi is anR-balanced function, then by using [7, Proposition 14.14], there is an exact uniqueN-homomorphismϕ:V⊗I→V Isatisfying the conditionϕ(v⊗i)=
ψ(v,i). SinceVisk-flat relative toRR, thenIV⊗RiIis injective. Ifϕ(Σvi⊗ai)=ϕ(Σvi⊗
a
i), thenθ(IV⊗RiI)(Σvi⊗ai)=θ(IV⊗RiI)(Σvi⊗ai). SinceθandIV⊗iI are injective,
thenΣ(vi⊗ai)=(Σvi⊗ai).
(2)⇒(1). Again consider the above diagram. LetIbe any left ideal ofRand letIV⊗R
iI(Σvi⊗ai)=IV⊗RiI(Σvi⊗ai), where Σvi⊗ai, Σvi⊗ai∈ V⊗RI. Let K1=ΣRai,
K2=ΣRai, andK=K1+K2. Nowθ(IV⊗iI)(Σvi⊗ai)=θ(IV⊗iI)(Σvi⊗ai), whence Σviai=Σviai. Now consider the following diagram, whereiK:K→I is the inclusion
map:
V⊗K
ϕK
IV⊗RiK
V⊗I
θ
V K iV K V .
(4.2)
By hypothesis,ϕKis monic. Thus,Σivi⊗ai=Σivi⊗aias an element ofV⊗K. Hence,
IV⊗RiK(Σiv⊗ai)=IV⊗iK(Σivi⊗ai)∈V⊗I, andΣvi⊗ai=Σvi⊗aias an element of
Proposition4.2. LetMbe a cancellable leftR-semimodule. ThenRRisMk-flat.
Proof. LetiK:K→M be the inclusion homomorphism. By [7, Proposition 14.16], R⊗RKKandR⊗RMM. Consider the following commutative diagram:
R⊗RK
IR⊗i
R⊗RM
K iK M,
(4.3)
sinceiKis injective, thenI⊗RiKis injective.
Corollary4.3. LetMbe a cancellable leftR-semimodule. Then every freeR
-semi-module isMk-flat.
Proof. The proof is immediate from Propositions2.3and4.2.
In module theory every projective module is flat. Now we see that this is true for certain special semimodules.
Proposition4.4. LetMbe a cancellable leftR-semimodule, whereRis a cancellative
completely subtractive semiring. Then everyk-regular projectiveR-semimodulePis Mk-flat.
Proof. By using [5, Theorem 19],P is isomorphic to a direct summand of a free
semimoduleF. ByCorollary 4.3,F isMk-flat. Hence, by usingProposition 2.3,PisMk -flat.
Corollary4.5. LetM be ak-regular leftR-semimodule andRa cancellative
com-pletely subtractive semiring. Then everyk-regular projectiveR-semimodulePisMk-flat.
Proof. We only need to show thatMis cancellable. SinceMisk-regular, then there
exists a freeR-semimoduleF such thatϕ:F→Mis surjective. Letm1+m=m2+m,
wherem1,m2,m∈M. Sinceϕis surjective, thenϕ(a1)+ϕ(a)=ϕ(a2)+ϕ(a), where
ϕ(a1)=m1, ϕ(a)=m, andϕ(a2)=m2. Since ϕ is k-regular, then a1+a+k1=
a2+a+k2, wherek1,k2∈Kerϕ. SinceF is cancellable, thena1+k1=a2+k2. Hence
ϕ(a1)=ϕ(a2).
Proposition4.6. LetMbe a cancellable leftR-semimodule. If Vis a freeR
-semimo-dule, then the following assertions hold: (a) VisMk-flat;
(b) V∗isM-injective.
Proof. By usingCorollary 4.3,VisMk-flat.
(i)⇒(ii). The proof is immediate fromProposition 3.3.
Acknowledgments. I wish to thank Dr. Michael H. Peel for his crucial comments.
References
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Current address: Department of Environmental Sciences and Mathematics, Faculty of Science and Health, University of East London, London, UK