Flat semimodules

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-semimodules_RM (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, Hom_R(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. LetU_{be 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: Hom_R(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 semimoduleV_Ris flat relative to a semimodule_RM(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 semimoduleV_Risk-flat relative to a semimodule_RM(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.

Proposition_2.3. _Let(V_α)_{α∈Abe an indexed set of right}R_{-semimodules. Then}⊕_AV_α

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 left}k_{-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. LetV_Rbe projective and_RMcancellable. 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-isomorphismsϕandϕ_{such that the following}

diagram commutes:

HomRM,HomN(U,T )

ϕ

Hom_R(α,IHom_N(U,T ))

HomRM,HomN

U,TN

ϕ

HomN

U⊗RM,T

Hom_N((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}

ϕ_Hom

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) Let}K_{be a subsemimodule of}M. 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:

Hom_RM,V∗

ϕ

HomR(iK,IV∗)

Hom_RK,V∗ ₀

ϕ

(V⊗M)∗ Hom(IV⊗iK,IC)

(V×K)∗ _0,

(3.4)

whereϕ_{andϕareN-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.

Proposition_4.1. _{The following statements about a cancellable right}R_-semimodule Vare equivalent:

(1) Visk-flat relative to_RR;

(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). Since}V_{is 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=ΣRa_i, 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. ThenR_RisMk-flat.

Proof. Leti_K: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.

Corollary_4.3. _LetM_{be a cancellable left}R_{-semimodule. Then every free}R

-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.

Proposition_4.4. _LetM_{be a cancellable left}R_{-semimodule, where}R_{is 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.

Corollary_4.5. _LetM _{be a}k_{-regular left}R_{-semimodule and}R_{a 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).

Proposition_4.6. _LetM_{be a cancellable left}R_{-semimodule. If} V_{is a free}R

-semimo-dule, then the following assertions hold: (a) VisMk-flat;

(b) V∗_{isM-injective.}

Proof. _{By usingCorollary 4.3,}V_isMk_-flat.

(i)⇒(ii). The proof is immediate fromProposition 3.3.

Acknowledgments. I wish to thank Dr. Michael H. Peel for his crucial comments.

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Current address: Department of Environmental Sciences and Mathematics, Faculty of Science and Health, University of East London, London, UK