Characterizations of projective and k projective semimodules

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CHARACTERIZATIONS OF PROJECTIVE

AND

k

-PROJECTIVE SEMIMODULES

HUDA MOHAMMED J. AL-THANI

Received 14 September 2001

To my late father

This paper deals with projective andk-projective semimodules. The results for projective semimodules are generalization of corresponding results for projective modules.

2000 Mathematics Subject Classification: 16Y60.

1. Introduction. Throughout this paper,Rdenotes a semiring with identity 1, all semimodulesMare leftR-semimodules and in all cases are unitary semimodules, that is, 1·m=mfor allm∈Mall leftR-semimodule_RM.

We recall here (cf. [1,2,3,4,5]) the following facts:

(a) letα:M→Nbe a homomorphism of semimodules. The subsemimodule Imα of N is defined as follows: Imα= {n∈N:n+α(m_{)=α(m)for somem,}

m_{∈M}. The homomorphismαis said to be an isomorphism ifαis injective} and surjective; to bei-regular ifα(M)=Imα; to bek-regular if form,m∈M, α(m)=α(m)implying thatm+k=m_{+kfor somek,k∈Kerα; and to be} regular if it is bothi-regular andk-regular;

(b) the sequenceK α→M β→N is called an exact sequence if Kerβ=Imα, and proper exact if Kerβ=α(K);

(c) for any two R-semimodules N, M, HomR(N,M):= {α : N →M |αis anR -homomorphism of semimodules}is a semigroup under addition. IfM, N,U, are R-semimodules and α: M → N is a homomorphism, then Hom(IU,α): HomR(U,M)→HomR(U,N) is given by Hom(IU,α)γ = αγ where IU is the identity;

(d) Pis a projective semimodule if and only if for each surjectiveR-homomorphism α:M→N, the induced homomorphism

¯

α: Hom(P,M) →Hom(P,N) (1.1)

is surjective;

(e) a leftR-semimodulePisMk-projective if and only if it is projective with respect to every surjectivek-regular homomorphismϕ:M→N.

InSection 2, we study the structure ofk-projective semimodules.Proposition 2.2

shows that for a semimoduleP, the class of all semimodulesM such thatP isMk -projective is closed under subtractive subsemimodules, factor semimodules, and gives a sufficient condition for the class to be closed undertaking homomorphic images.

module theory and semimodule theory. InSection 3, we characterize projective and k-projective semimodules via the Hom functor. Theorems3.5and3.7assert thatPis M-projective (Mk-projective) if and only if Hom_R(P,−)preserves the exactness of all proper exact sequencesM α_→M β_{→M, withβ k-regular (bothαandβ k-regular).}

2. k-projective semimodules. We study the structure ofk-projective semimodules via the Hom function. We show that the class of all semimodulesM, such thatP is Mk-projective, is closed under subtractive subsemimodules, factor semimodule and undertaking homomorphic image for ak-regular homomorphism.

For provingProposition 2.2we need the following proposition, which is modified from [5, Theorem 2.6].

Proposition_2.1. _LetR_{be a semiring,}

(i) if0→M α→M β_{→M is a proper exact sequence ofR-semimodules andαis}

k-regular, then for everyR-semimodule E,

0→Hom_R(E,M) α¯→Hom_RE,M β¯_{→HomE,M (2.1)}

is a proper exact sequence of Abelian semigroups andα¯is regular, whereα(ξ)¯ =

αξforξ∈Hom(E,M)andβ(γ)¯ =βγforγ∈Hom(E,M_);

(ii) ifM α→M β_{→M→0is a proper exact sequence ofR-semimodules andβis}

k-regular, then for everyR-semimoduleE,

0 →HomM_,E β¯_→HomM,E α¯_{→Hom(M,E) (2.2)}

is a proper exact sequence of Abelian semigroups andβ¯is regular, whereβ(ξ)¯ =

βξ.

Proof. _{(i) Since the sequence 0}→M α→M β→M _{is proper exact, then the}

se-quence is exact withαbeingi-regular. Using [5, Theorem 2.6], the sequence

0 →Hom(E,M) α¯→HomE,M β¯_{→HomE,M (2.3)}

is exact with ¯αbeing regular. This means that the sequence is proper exact. (ii) can be proved by the same argument.

Proposition_2.2. _LetP _{be a left}R-semimodule. If₀→M θ→M η→M→_{0is a} proper exact sequence withθbeing regular,ηbeingk-regular, andPisMk-projective, thenP isk-projective relative to bothM_andM.

Proof. _Let Ψ _:M →N _{be surjective} k_{-regular homomorphism and} α_:P →N

be homomorphism. Sinceηis surjective k-regular, thenΨηis k-regular. SinceP is Mk-projective, then there exists a homomorphismϕ:P→Msuch that the following diagram commutative:

P

α ϕ

M η M Ψ _{N 0}

(2.4)

To prove thatP isM_{k-projective. LetΨ:M→Nbe a surjectivek-regular} homo-morphism and setK=KerΨ. Since Ψ is surjective k-regular homomorphism, then M_{/K N. Define ¯θ:M/K→M/θ(K)by the rule ¯θ(m/K)=θ(m)/θ(K), and ¯η:}

M/θ(K)→M_{by the rule ¯η(m/θ(K))=η(m). Clearly, both ¯θand ¯ηare well defined} homomorphisms. Consider the sequence 0→ M_/K θ¯_{→ M/h(K)} η¯

→ M _{→ 0. Let}

m/θ(K)∈Ker ¯η, thenη(m)=0, hencem∈Kerη=θ(M_{). Hence Ker ¯η=θ(M}¯ _/K). Clearly ¯ηis surjective, and ¯θis injective. Sinceθisi-regular, then ¯θisi-regular. Now consider the following commutative diagram:

0

0 M

πk

θ

M

πh(k) η

M ₀

0 M/K θ¯ M/θ(K) η¯ M ₀

0 0 0

(2.5)

Applying HomR(P,−)to this diagram we have the commutative diagram

0

0 HomP,M

(πk)∗

θ∗

Hom(P,M)

(πθ(k))∗

η∗

HomP,M

I∗

0 HomP,M_/K θ∗¯ _HomP,M/θ(K) η∗¯ _Hom(P,M)

0 0

(2.6)

UsingProposition 2.1, and sinceP is Mk-projective, then all rows and columns are proper exact sequence. We should show that(πK)∗is surjective. Letα∈Hom(P,M/K). Since(πθ(K))∗is surjective, then there existsβ∈Hom(P,M)such that(πθ(K))∗(β)=

¯

θ∗(α). Now ¯η∗θ¯∗(α)=η¯∗((πθ(K))∗(β)) =I∗η∗(β)=0. Hence η∗(β)⊂KerI∗=0. Henceβ=θ∗(γ)whereγ∈Hom(P,M). Thus ¯θ∗(α)=(πθ(K))∗(β)=(πθ(K))∗θ∗(γ)=

¯

θ∗(πK)∗(γ). Again by Proposition 2.1, ¯θ∗ is injective, hence α= (πK)∗(γ). Thus

(πK)∗is surjective. ThereforeP isMk-projective.

give us a sufficient condition to be closed undertaking a homomorphic image. Since for every subsemimoduleKofM, the canonical surjectionπK:M→M/Kisk-regular surjective, then the classΩ(P)is closed under factor semimodules.

We know that in module theory any projective module is a direct summand of a free module. However, for arbitary semirings this is not true.

Example_2.3. _LetR_{be the field}Z/p_{for any prime integer. Let}S= {{θ¯},Z/p}

setR_{= {(a,I)¯ : ¯a∈I∈S}, that is}

R₌¯_0,¯_0,a,Z/¯ _p,a¯_∈Z/p. (2.7)

Define operations⊕and⊗onR_{by setting}

¯

a,I⊕_b,H¯ _=a¯+¯_b,I+H,

¯

a,¯_I⊗b,H¯ _=a¯b,IH¯ _. (2.8)

ClearlyR_{is semiring. LetI}+_{(R)be the set of all additively idempotent elements of}

R_:I+_{(R)= {(}¯_0,{¯_0}),(0¯_{,Z/p)}. We note that the functionα:R→I}+_{(R), defined} byα(¯0,{¯0})=(¯0,{¯0})andα(a,¯Z/p)=(0¯,Z/p), is a surjectiveR_{-homomorphism} of left R_{-semimodules. Furthermore, the restriction of α toI}+_{(R) is the identity} map. ThereforeI+_{(R)is a retract ofR. SinceRis projective, as a left semimodule} over itself, by [5, Corollary 15.13] we see thatI+_{(R)is also projective. IfI}+_{(R)is a} free semimodule, then(¯_{0,Z/p)is a basis, but(a,Z/¯ p)(}¯_0,Z/p)=(¯_0,Z/p)for every ¯a∈Z/p. HenceI+_{(R)is not free. Now, suppose thatI}+_{(R)is a direct} sum-mand of a freeR_{-semimodule, sayF. Then F K⊕I}+_{(R) for anR-semimodule}

K. Let ϕ: K⊕I+_{(R)→ F = ⊕αR}

α be an isomorphism, where Rα = R for all α. Since(0,(¯_{0,Z/p))is an idempotent element where 0∈K, thenϕ(0,(}¯_0,Z/p))is an idempotent element in F. Since the only idempotent elements ofR _are(¯0,{¯0}) and (¯_{0,Z/p), then ϕ(0,(0}¯_{,Z/p))=(xα), where only finite numbers of xα are} nonzeros and xα =(¯0,Z/p). Let (yα) have components yα = (¯1,Z/p) for α, with xα=(¯0,Z/p), and otherwiseyα=(¯0,{¯0}). Suppose ϕ(k,(¯0,Z/p))=(ya), where 0≠k∈K. Clearly,p(yα)=(xa),pϕ(k,(¯0,Z/p))=ϕ(0,(¯0,Z/p)). Hence

p(k,(¯_0,Z/p))=(0,(¯_{0,Z/p)). Hence pk=0. Therefore p(k,(}¯_0,{¯_{0}))=0. Hence} (k,(¯_0,{¯_{0}))has an additive inverse. Thus,ϕ(k,(}¯_0,{¯_{0}))also has an additive inverse.} Now, every element ofF is of the form(uα), uα∈R. Since every nonzero element of

R_{has no additive inverse, then every nonzero element ofF has no additive inverse.} Thus we have a contradiction. Therefore,I+(R)is not a direct summand of a free R_-semimodule.

3. Characterizations of projective andk-projective semimodules. We character-ize projective andk-projective semimodules via the Hom functor.

Lemma 3.1. let R be a semiring and letM α→M β→ M be a sequence of R -semimodules. Then, the sequence is exact if there exists a commutative diagram

0 0

N

η

M

Ψ α

M

ϕ β

M

K

θ

0 0

(3.1)

ofR-semimodule in which thenonhorizontalsequences are all exact.

Proof. Letx∈Kerβ, thenηϕ(x)=β(x)=0, henceϕ(x)∈Kerη. Since Kerη=0,

thenx∈Kerϕ=Imθ. Hencex+θ(kl)=θ(k2), thereforex+θ(Ψ(ml))=θ(Ψ(m2)). SinceθΨ=α, thenx+α(m1)=α(m2). Thusx∈Imα. Conversely, letx∈Imα, then x+α(m1)=α(m2)for somem1,m2∈M. Again sinceθΨ=α, thenx+θΨ(m1)=

θΨ(m2), hence x∈Imθ. But Imθ=Kerϕ, hence β(x)=ηϕ(x)=0. Thus Imα= Kerβ.

Since every proper exact sequence is exact sequence. Then we have the following corollary.

Corollary 3.2. LetR be a semiring and letM α→M β→M be a sequence of R-semimodules. Then the sequence is proper exact if there exists a commutative dia-gram

0 0

N

η

M

Ψ α

M

ϕ β

M

K

θ

0 0

(3.2)

Proof. Since every proper exact sequence is exact sequence, then usingLemma

3.1, we have Kerβ= Imα, hence α(M_{) ⊂ Kerβ. Now let x ∈ Kerβ, thenβ(x) =}

ηϕ(x)=0, hence ϕ(x)∈Kerη. But Kerη=0, thereforex∈Kerϕ=θ(K). Hence x=θ(k)=θ(Ψ(m_{))=α(m). Thusα(M)=Kerβ.}

Corollary _3.3. _LetR _{be a semiring and let}M α→M β→M _{be a sequence of} R-semimodules withβbeingk-regular. Then the sequence is proper exact if and only if there exists a commutative diagram

0 0

N

η

M

Ψ α

M

ϕ β

M

K

θ

0 0

(3.3)

ofR-semimodules in which thenonhorizontalsequences are all proper exact.

Proof. _LetM α→M β→M_{be a proper exact sequence with}β_beingk_-regular.

Consider the following diagram:

0 0

M/Kerβ η

M

Ψ α

M

ϕ β

M

Kerβ

i

0 0

(3.4)

whereΨ(m_{)=α(m)for allm∈M,i(x)=x for allx∈Kerβ,ϕ(m)=m/Kerβ} for allm∈M, andη(m/Kerβ)=β(m)for allm/Kerβ∈M/Kerβ. Letm1/Kerβ=

m2/Kerβ, thenm1+k1=m2+k2,k1,k2∈Kerβ. Henceβ(m1)=β(m2), thereforeη is well defined. Now ifβ(m1)=β(m2), and sinceβisk-regular, we havem1+k1=

Hence m1/Kerβ=m2/Kerβ, therefore ηis injective. Clearly the sequence 0→ Kerβ i→M ϕ→M/Kerβ→0 is proper exact sequence. Thus diagram (3.4) is a com-mutative diagram in which the nonhorizontal sequences are all proper exact.

Conversely, seeCorollary 3.2.

Corollary _3.4. _LetR _{be a semiring and let}M α→M β→M _{be a sequence of} R-semimodules withβbeingk-regular. Then the sequence is exact if and only if there exists a commutative diagram

0 0

N

η

M

Ψ α

M

ϕ β

M

K

θ

0 0

(3.5)

ofR-semimodules in which thenonhorizontalsequences are all exact.

Proof. _LetM α→M β→M_{be an exact sequence with}β_beingk_{-regular. Consider}

the following diagram:

0 0

M/Imα

η

M

Ψ α

M

ϕ β

M

Imα

i

0 0

(3.6)

whereΨ(m_{)=α(m)∈Imαfor all m∈M, i(x)=x for allx ∈Imα, ϕ(m)=}

m/Imαfor allm∈M, andη(m/Imα)=β(m)for allm/Imα∈M/Imα. Letm1/Imα

injective. Clearly, the sequence 0→Imα i→M ϕ→M/Imα→0 is exact. Thus diagram (3.6) is a commutative diagram in which the nonhorizontal sequences are all exact.

Conversely, seeLemma 3.1.

Theorem_3.5. _{The following statements about left}R-semimoduleP_{are equivalent:} (i) Pis projective;

(ii) for every proper exact sequence of leftR-semimodulesM α_→M β_→Mwithβ

beingk-regular the sequence

HomP,M α¯_{→Hom(P,M) β}¯

→HomP,M _(3.7)

is proper exact.

Proof. (ii)⇒(i). Letα:N→M be surjective homomorphism. SinceN α→M→0 is

proper exact sequence withM→0 being regular, then by (ii) Hom(P,N)→Hom(P,M)→ 0 is proper exact. ThereforePis projective.

(i)⇒(ii). LetPbe a projective semimodule. Suppose thatM α_→M β_{→Mis proper} exact withβbeingk-regular. Consider the sequence 0→Kerβ →i M π→M/Kerβ→0, whereπ(m)=m/Kerβis the canonical surjection. Clearlyiis injective. SinceP is projective, and using [1, Theorem 10], the sequence

0 →Hom_R(P,Kerβ) ¯i→Hom(P,M) π¯→Hom(P,M/Kerβ)→0 (3.8)

is proper exact sequence. DefineΨ:M_{→Kerβandη:M/Kerβ→M, whereΨ(m)=}

α(m_{)andη(m/Kerβ)=β(m). Letη(m/Kerβ)=η(m/Kerβ), thenβ(m)=β(m).} Since βis k-regular, then m+k=m_{+k, where k,k ∈Kerβ. Hence m/Kerβ=}

m_{/Kerβ. Thereforeηis injective. Now consider the commutative diagram}

0 0

HomR(P,M/Kerβ)

¯

η

Hom_RP,M

¯ Ψ ¯

α

HomR(P,M)

¯

π

¯

β

Hom_RP,M

HomR(P,Kerβ)

¯

i

0 0

(3.9)

where ¯Ψ(ξ)=Ψξand ¯η(γ)=ηγ,ξ∈Hom_R(P,M_)andγ∈Hom

following diagram is commutative:

P

θ ξ

M

Ψ Kerβ 0

(3.10)

Thus ¯Ψ is surjective. Thus the nonhorizontal sequences are all proper exact. Using

Corollary 3.2, the sequence

HomRP,M α¯→HomR(P,M)

¯

β

→HomRP,M (3.11)

is proper exact.

Corollary_3.6. _{The following statement about left}R_-semimoduleP_{are equivalent:} (i) Pis projective;

(ii) for every proper exact sequence of leftR-semimodulesM α_→M β_→Mwithβ

being regular, the sequence

HomP,M α¯

→Hom(P,M) β¯→HomP,M _(3.12)

is proper exact.

Proof. _{It is a consequence ofTheorem 3.5.}

Theorem_3.7. _{The following statements about left}R-semimoduleP_{are equivalent:} (i) Pisk-projective;

(ii) for every proper exact sequence of leftR-semimodulesM α→M β→M with bothαandβbeingk-regular, the sequence

HomP,M α¯_{→Hom(P,M) β}¯

→HomP,M _(3.13)

is proper exact.

Proof. _(ii)⇒_{(i). Let} α_: N→ M →_{0 be} k_{-surjective homomorphism. Since} N→ M →0 is proper exact sequence withα being k-regular, then by (ii) Hom(P,N)→ Hom(P,M)→0 is proper exact. ThereforePisk-projective.

(i)⇒(ii). It is similar to the proof ofTheorem 3.5and using [2, Theorem 8].

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

Thanks are also directed to Dr Mustafa A. Mustafa for his help.

References

[1] H. M. J. Al-Thani,A note on projective semimodules, Kobe J. Math.12(1995), no. 2, 89–94. [2] ,k-projective semimodules, Kobe J. Math.13(1996), no. 1, 49–59.

[4] M. Takahashi,On the bordism categories. II. Elementary properties of semimodules, Math. Sem. Notes Kobe Univ.9(1981), no. 2, 495–530.

[5] ,On the bordism categories. III. FunctorsHomand⊗for semimodules, Math. Sem. Notes Kobe Univ.10(1982), no. 1, 211–236.