Bi-amalgamations subject to the arithmetical property

0.14 MB·

by  S. Kabbaj

BI-AMALGAMATIONS SUBJECT TO THE ARITHMETICAL PROPERTY (⋆)
6 S.KABBAJ(1),N.MAHDOU,ANDM.A.S.MOUTUI
1
0
2 Abstract. This paper establishes necessary and sufficient conditions for a bi-
amalgamationtoinheritthearithmeticalproperty,withapplicationsontheweak
n
global dimensionandtransferofthesemihereditaryproperty. Thenewresults
a
comparetopreviousworkscarriedonvarioussettingsofduplicationsandamal-
J
gamations,andcapitalizeonrecentresultsonbi-amalgamations. Allresultsare
8 backedwithnewandillustrativeexamplesarisingasbi-amalgamations.
2

]
C
A
. 1. Introduction
h
t
a Throughout, allringsconsideredarecommutativewithunityandallmodules
m
are unital. Let f :A→B and g:A→C be two ring homomorphisms and let J
[
and J′ be two idealsof B and C, respectively, such thatI := f−1(J)=g−1(J′). The
o
1 bi-amalgamation(orbi-amalgamatedalgebra)ofAwith(B,C)along(J,J′)withrespect
v
3 to(f,g)isthesubringofB×Cgivenby
5
6 A⊲⊳f,g(J,J′):= (f(a)+j,g(a)+j′)|a∈A,(j,j′)∈J×J′ .
7
n o
0
. Thisconstructionwasintroducedin[26]asanaturalgeneralizationofduplications
1
0 [12, 14, 17, 18, 27, 30] and amalgamations [15, 16, 19]. In [26], the authors pro-
6
videoriginalexamplesofbi-amalgamationsand,inparticular,showthatBoisen-
1
: Sheldon’sCPI-extensions[7]canbeviewedasbi-amalgamations(Noticethat[15,
v
i Example2.7]showsthatCPI-extensionscanbeviewedasquotientringsofamalga-
X
matedalgebras). Theyalsoshowhoweverybi-amalgamationcanariseasanatural
r
a pullback(orevenasaconductorsquare)andthencharacterizepullbacksthatcan
ariseasbi-amalgamations. ThisallowedthemtocharacterizeTraverso’sglueings
ofprimeideals[29,31,32,33]whichcanbeviewedasspecialbi-amalgamations.
Then,thelasttwosectionsdeal,respectively,withthetransferofsomebasicring
theoreticpropertiestobi-amalgamationsandthestudyoftheirprimeidealstruc-
tures. Alltheirresultsrecoverknownresultsonduplicationsandamalgamations.
Date:January29,2016.
2010Mathematics SubjectClassification. 13F05, 13A15, 13E05, 13F20, 13C10, 13C11, 13F30, 13D05,
16D40,16E10,16E60.
(⋆)SupportedbyKingFahdUniversityofPetroleum&MineralsunderResearchGrant#RG1310.
(1)Correspondingauthor.
1
2 S.KABBAJ,N.MAHDOU,ANDM.A.S.MOUTUI
Finally,itisworthwilerecallingthat,veryrecently,Finocchiaroinvestigatedneces-
saryandsufficientconditionsforanamalgamatedalgebratoinheritvariousPru¨fer
conditions(includingthearithmeticalproperty)[19].
Thispaperstudiesthetransferofthearithmeticalpropertyandrelatednotions
to bi-amalgamations. A ring R is arithmetical if every finitely generated ideal is
locallyprincipal[20,25];andRissemihereditaryifeveryfinitelygeneratedidealis
projective [11]. The following diagram of implications summarizes the relations
betweenthemainthreenotionsinvolvedinthispaper:
Rissemihereditary
⇓
w.dim(R)≤1
⇓
Risarithmetical
where w.dim(R) denotes the weak global dimension of R. Recall that all these
propertiesare identical to the notion of Pru¨fer domain if R has no zero-divisors,
andthattheaboveimplicationsareirreversible,ingeneral,asshownbyexamples
provided in [1, 4, 5, 6, 8, 9, 10, 12, 13, 22, 23, 24]. Very recently, these conditions
(amongotherPru¨ferconditions)werethoroughlyinvestigatedinvariouscontexts
ofduplications[12].
Thispaperestablishesnecessaryandsufficientconditionsforabi-amalgamation
toinheritthe arithmeticalproperty,withapplicationsonthe weakglobaldimen-
sionandtransferofthesemihereditaryproperty. Section2isdevotedtothetransfer
results,themainofwhich(Theorem2.1)statesthat”A⊲⊳f,g(J,J′)isarithmeticalifand
onlyifboth f(A)+Jandg(A)+J′arearithmeticaland,foreverym∈Max(A,I ),J =0
o Sm
orJ′ =0”. Followseveralapplicationsfeaturingthetransferofotherrelatedprop-
S′m
ertiesinbi-amalgamations,amalgamations,andduplications. Allobtainedresults
recoverandcomparetopreviousworkscarriedonvarioussettingsofduplications
andamalgamations,andcapitalizeonrecentresultsonbi-amalgamations. (Asto
previousworksonpullbacks,seeRemark2.2.) InSection3,allresultsarebacked
withnewandillustrativeexamplesarisingasbi-amalgamations.
Notice,atthispoint,thatinthepresenceoftheequality f−1(J)=g−1(J′), J=Bif
andonlyifJ′=C;andinthiscaseA⊲⊳f,g(J,J′)=B×C. Therefore,inthispaper,we
willomit this case (i.e., J and J′ willalwaysbe proper)since itis known thatthe
abovePru¨fernotionsarestableunderfiniteproducts(cf. [12,p. 251]).
Throughout,foraringR,Spec(R)(resp.,Max(R))willdenotethesetofallprime
(resp., maximal)idealsof R, and, for anyidealI of R, Spec(R,I) (resp., Max(R,I))
willdenotethesetofallprime(resp.,maximal)idealsofRcontainingI.
BI-AMALGAMATIONSSUBJECTTOTHEARITHMETICALPROPERTY 3
2. Results
Let f :A→Band g:A→Cbetworinghomomorphisms andlet J and J′ betwo
properidealsofBandC,respectively,suchthatI := f−1(J)=g−1(J′). Allalongthis
o
section, A⊲⊳f,g(J,J′) will denote the bi-amalgamation of A with (B,C) along (J,J′)
withrespectto(f,g). Thissectioninvestigatesthearithmeticalandsemihereditary
propertiesaswellastheweakglobaldimensioninbi-amalgamations.
The first main result establishes necessary and sufficient conditions for a bi-
amalgamation to inherit the arithmetical property. To this purpose, let us adopt
thefollowingnotation:
Foranyp∈Spec(A,I )(resp.,∈Max(A,I )),considerthemultiplicativesubsets
o o
S := f(A−p)+J and S′ :=g(A−p)+J′
p p
ofBandC,respectively,andlet
fp:Ap→BSp and gp:Ap→CS′p
bethecanonicalringhomomorphisms inducedby f and g. Onecaneasilycheck
that
f−1(J )=g−1(J′ )=(I ) .
p Sp p S′p o p
Moreover,by[26,Lemma5.1],P:=p⊲⊳f,g(J,J′)isaprime(resp.,maximal)idealof
A⊲⊳f,g(J,J′)and,by[26,Proposition5.7],wehave
A⊲⊳f,g(J,J′) (cid:27)A ⊲⊳fp,gp (J ,J′ ).
P p Sp S′p
(cid:16) (cid:17)
Thesefactswillbeusedinthesequelwithoutexplicitreference. Recallthatalocal
arithmetical ring is also called a chained ring (i.e., its lattice of ideals is totally
orderedbyinclusion).
Theorem2.1. Undertheabovenotation,wehave:
(1) A⊲⊳f,g(J,J′)isachainedringifandonlyifboth f(A)+Jandg(A)+J′arechained
ringsandJ=0orJ′=0.
(2) A⊲⊳f,g(J,J′)isarithmeticalifandonlyifboth f(A)+Jandg(A)+J′arearithmetical
and,foreverym∈Max(A,I ),J =0orJ′ =0.
o Sm S′m
Proof. (1) By [26, Proposition 3.1], A⊲⊳f,g (J,J′) arises as a pullback D:=α× β
A
withKer(α)=J,Ker(β)=J′,andp (D)= f(A)+J(resp.,p (D)=g(A)+J′)whereIop
1 2 1
(resp.,p )denotestherestrictiontoDoftheprojectionof(f(A)+J)×(g(A)+J′)into
2
f(A)+J (resp., g(A)+J′). Moreover, recall that the chained ring notion is stable
underfactorringand f(A)+J (cid:27) g(A)+J′ [26,Proposition4.1(3)]). Therefore,theresult
J J′
followsreadilyfrom[19,Proposition4.9].
(2) First note that (1) is the local version of (2). In order to see this, recall
that A⊲⊳f,g (J,J′) is local if and only if both f(A)+J and g(A)+J′ are local; and
4 S.KABBAJ,N.MAHDOU,ANDM.A.S.MOUTUI
m⊲⊳f,g (J,J′) is the maximal ideal of A⊲⊳f,g(J,J′), where m is the unique maximal
idealofAcontainingI [26,Proposition5.4]. Moreover,inviewoftheisomorphism
o
A⊲⊳f,g(J,J′)
(cid:27) f(A)+J
0×J′
givenby[26,Proposition4.1(2)],wededucethat
m⊲⊳f,g(J,J′)
≃ f(m)+J
0×J′
is the maximal ideal of f(A)+J and, similarly, g(m)+J′ is the maximal ideal of
g(A)+J′. HenceSm andS′m shallconsistofunitsonlysothat JSm =J and JS′′m =J′,
asdesired.
Now,assumethatA⊲⊳f,g(J,J′)isarithmetical. Letm∈Max(A,I )andM:=m⊲⊳f,g
o
(J,J′). Therefore
Am⊲⊳fm,gm (JSm,JS′′m)(cid:27) A⊲⊳f,g(J,J′) M
(cid:16) (cid:17)
isachainedring(sincethearithmeticalpropertyisstableunderlocalization). By
(1),J orJ′ isnull,asdesired. Next,letL∈Spec(f(A)+J)andconsidertheprime
Sm S′m
idealofA⊲⊳f,g(J,J′)givenby
L:= L×(g(A)+J′) ∩ A⊲⊳f,g(J,J′) .
(cid:16) (cid:17) (cid:16) (cid:17)
IfJ*L,thenby[26,Proposition5.3(2)],
(f(A)+J) (cid:27) A⊲⊳f,g(J,J′)
L
L
(cid:16) (cid:17)
isarithmetical. Next,assumethatJ⊆L. By[26,Lemma5.2],wehave
L:=p⊲⊳f,g(J,J′)
wherep:= f−1(L)∈Spec(A,I )andonecaneasilyverifythatL= f(p)+J. So,
o
A ⊲⊳fp,gp (J ,J′ )(cid:27) A⊲⊳f,g(J,J′)
p Sp S′p L
(cid:16) (cid:17)
is a chained ring. By (1), f (A )+J is a chained ring and (as seen above) with
p p Sp
maximalideal f (pA )+J . Thefactthat f (A )+J islocalyields:
p p Sp p p Sp
Claim1. f (A )+J =(f(A)+J) .
p p Sp L
Indeed,firstobservethat
S =(f(A)+J)\(f(p)+J)=(f(A)+J)\L
p
and, hence, both f (A )+J and (f(A)+J) are subrings of B . The forward
p p Sp L Sp
inclusionisobvious. Toprovetheother,letx∈(f(A)+J) ;thatis,
L
f(a)+i 1 f(a) i
x= = +
f(s)+j f(s)+j! 1 ! f(s)+j
forsomea∈A,s∈A\pandi,j∈J. Clearly,itsufficestoshowthat
1
∈ f (A )+J .
f(s)+j p p Sp
BI-AMALGAMATIONSSUBJECTTOTHEARITHMETICALPROPERTY 5
Thisistruesinceonecancheckthat
f(s)+j f(s) j
= + < f (pA )+J
1 1 1 p p Sp
as f (pA )+J isthemaximalidealof f (A )+J ,provingtheclaim.
p p Sp p p Sp
By (1), (f(A)+J) is arithmetical. Consequently, f(A)+J is (locally) arithmetical
L
andsoisg(A)+J′viasimilararguments.
Conversely,assume f(A)+Jandg(A)+J′arearithmeticaland,∀m∈Max(A,I ),
o
J or J′ is null. Let M∈Max(A⊲⊳f,g (J,J′)). Suppose that J×J′ *M. By [26,
Sm S′m
Proposition5.3(2)],thereisL,say,inSpec(f(A)+J)suchthat
(A⊲⊳f,g(J,J′))M(cid:27)(f(A)+J)L.
So,inthiscase,(A⊲⊳f,g(J,J′))Misobviouslyarithmetical. Next,supposeJ×J′⊆M.
By[26,Proposition5.3(1)&Lemma5.1],thereisauniquem∈Max(A,I )suchthat
o
M=m⊲⊳f,g(J,J′).
Byhypothesiswehave,say,J′ =0. Now,letL:= f(m)+J,aprimeidealof f(A)+J.
S′m
Itfollows,via[26,Proposition4.1(2)]andClaim1,that:
A⊲⊳f,g(J,J′) M (cid:27) Am⊲⊳fm,gm (JSm,0)
(cid:16) (cid:17) (cid:27) fm(Am)+JSm
= (f(A)+J) .
L
So, in this case too, (A⊲⊳f,g (J,J′))M is arithmetical. Consequently, A⊲⊳f,g (J,J′) is
arithmetical,completingtheproofofthetheorem. (cid:3)
Remark 2.2. In Proposition 3.2 of [26], it is proved that every bi-amalgamation
A⊲⊳f,g(J,J′) can be viewed as a conductor square with conductor J×J′. Boynton
examined the transfer of the arithmetical property to conductor squares in the
special case where the conductor ideal is regular [8, Theorem 3.3] (and also [9,
Theorem 4.1]). We cannot appeal to this result in the context of Theorem 2.1
since, under the assumption “J×J′ is regular,” the bi-amalgamation A⊲⊳f,g (J,J′)
can never satisfy the arithmetical property (because of the necessary condition:
∀m∈Max(A,I ), J =0or J′ =0). Thisremarkisalsovalidfortheforthcoming
o Sm S′m
Corollary 2.8 (on the weak global dimension) and Corollary 2.11 (on the semi-
hereditaryproperty).
Remark2.3. ObservethatTheorem2.1(1)canalsoreadasfollows: A⊲⊳f,g(J,J′)isa
chainedringifandonlyif“J=0and g(A)+J′isachainedring”or“J′=0and f(A)+J
is a chained ring” which is obvious from the facts that the chained ring notion is
stableunderfactorringand f(A)+J (cid:27) g(A)+J′ [26,Proposition4.1(3)]).
J J′
As an illustrative example for Theorem 2.1, Example 3.1 provides an original
arithmeticalringwhicharisesasabi-amalgamation.
6 S.KABBAJ,N.MAHDOU,ANDM.A.S.MOUTUI
RecallthattheamalgamationofAwithBalongJwithrespectto f isgivenby
A⊲⊳f J:= (a,f(a)+j)|a∈A,j∈J .
n o
Clearly, every amalgamation can be viewed as a special bi-amalgamation, since
A⊲⊳f J=A⊲⊳idA,f (f−1(J),J). Accordingly, Theorem 2.1 covers the special case of
amalgamations,asrecordedbelow.
Corollary2.4. Undertheabovenotation,wehave:
(1) A⊲⊳f J isachainedringifandonlyifbothAand f(A)+Jarechainedringsand
J=0or f−1(J)=0.
(2) A⊲⊳f JisarithmeticalifandonlyifbothAand f(A)+Jarearithmeticaland,for
everym∈Max(A,f−1(J)),JSm =0or fm−1 JSm =0.
(cid:0) (cid:1)
Remark2.5. Recently,Finocchiaroprovedthefollowingresultforthetransferofthe
arithmeticalpropertytoamalgamations: ”Assumethat,foreachm∈Max(A,f−1(J)),
either fm is surjective or fm−1 JSm ,0. Then, A⊲⊳f J is arithmetical if and only if A is
arithmetical,J =0foreachm(cid:0)∈M(cid:1)ax(A,f−1(J)),andforanym′∈Max(B)notcontaining
Sm
J,theidealsofBm′ aretotallyorderedbyinclusion”[19,Proposition4.10]. Corollary2.4
covers this result due to the basic fact that if fm is surjective and JSm ,0 then
fm−1 JSm ,0; combined with the two-type maximal ideal structure of A⊲⊳f J (cf.
[16,(cid:0)Pro(cid:1)position 2.6]and[19, Proposition 2.5]);precisely,Am(cid:27)(A⊲⊳f J)m⊲⊳fJ when
JSm =0andBm′ (cid:27)(A⊲⊳f J)m′ wherem′= (a,f(a)+j)∈A⊲⊳f J| f(a)+j∈m′ .
n o
Remark 2.6. Assume J,0. Then Remark 2.3 combined with [26, Corollary 4.6]
yield: A⊲⊳f J is a chained ring (resp., valuation domain) if and only if f−1(J)=0 and
f(A)+Jisachainedring(resp.,valuationdomain).
For an original example of arithmetical ring arising as an amalgamation, see
Example3.2. Next,letI beaproperidealofA. The(amalgamated)duplicationof
AalongIisaspecialamalgamationgivenby
A⊲⊳I:=A⊲⊳idA I= (a,a+i)|a∈A,i∈I .
n o
The above corollary recovers known results on the transfer of the arithmetical
propertytoduplications,asshownbelow.
Corollary2.7([12,Theorem3.2(1)&Corollary3.8(1)]). Wehave:
(1) A⊲⊳IisachainedringifandonlyifAisachainedringandI=0.
(2) A⊲⊳IisarithmeticalifandonlyifAisarithmeticalandIm=0,∀m∈Max(A,I).
Asanother applicationof Theorem2.1, we getnecessaryandsufficient condi-
tions for a bi-amalgamation to have weak global dimension at most 1. For this
purpose,letNil(R)denotethenilradicalofaringR.
BI-AMALGAMATIONSSUBJECTTOTHEARITHMETICALPROPERTY 7
Corollary 2.8. Assume w.dim(f(A)+J)≤1, w.dim(g(A)+J′)≤1, J∩Nil(B)=0,
J′∩Nil(C)=0and,∀m∈Max(A,I ),J =0orJ′ =0. Thenw.dim A⊲⊳f,g(J,J′) ≤1.
o Sm S′m
TheconverseholdsifI isradical. (cid:16) (cid:17)
o
Proof. Recall that a ring R has weak global dimension at most 1 if and only if
R is arithmetical and reduced [5, Theorem 3.5]. A combination of this fact with
Theorem 2.1 and [26, Proposition 4.7] (on the transfer of the reduced property)
leadstotheconclusion. (cid:3)
The converseof Corollary2.8isnottrue ingeneral. Acounter-exampleinthe
special case of amalgamations is given in Example 3.3. Also, as an illustrative
example for this result, Example 3.4 features an original example of a ring with
weakglobaldimension≤1whicharisesasabi-amalgamation.
Forthespecialcaseofamalgamations,wegetamorewell-roundedresult:
Corollary2.9. Undertheabovenotation,wehave:
w.dim(A⊲⊳f J)≤1ifandonlyifw.dim(A)≤1, f(A)+Jisarithmetical(resp.,Gaussian),
J∩Nil(B)=0and,foreverym∈Max(A,f−1(J)),JSm =0or fm−1 JSm =0.
(cid:0) (cid:1)
Proof. CombineCorollary2.4(2)and[26,Corollary4.9]withthewell-knownfacts
thatw.dim(R)≤1ifandonlyifRisreducedandarithmetical(resp.,reducedand
Gaussian)[5,Theorems3.5&4.8]. (cid:3)
Remark2.5isalsovalidforCorollary2.9and[19,Proposition4.11]ontheweak
globaldimension. SeeExample3.3wherew.dim(A⊲⊳f J)≤1andw.dim(f(A)+J)>
1. Corollary2.9recoversaknownresultforduplications:
Corollary2.10([12,Theorem4.1(1)]). Wehave:
w.dim(A⊲⊳I)≤1ifandonlyifw.dim(A)≤1andIm=0, ∀ m∈Max(A,I).
A ring R is semihereditary if and only if R is coherent and w.dim(R)≤1 [5,
Theorem3.3]. A combination of this factwith Corollary2.8and [26, Proposition
4.2]establishesthetransferofthesemihereditarypropertytobi-amalgamationsin
thespecialcaseofNoetheriansettings.
Corollary2.11. Assumethat f(A)+Jandg(A)+J′areNoetheriansemihereditaryrings,
J∩Nil(B)=0,J′∩Nil(C)=0and,∀m∈Max(A,I ),J =0orJ′ =0. Then,A⊲⊳f,g(J,J′)
o Sm S′m
isaNoetheriansemihereditaryring. TheconverseholdsifI isradical.
o
For the special case of amalgamations, we have a more well-rounded general
result. For this purpose, we first recall the following result which examines the
transferofcoherencetoamalgamations.
8 S.KABBAJ,N.MAHDOU,ANDM.A.S.MOUTUI
Lemma2.12([2, Theorem2.2]). Supposethat f−1(J)and J arefinitelygenerated inA
and f(A)+J, respectively. Then: A⊲⊳f J is coherent if and only if A and f(A)+J are
coherent.
Contrast this result with [19, Proposition 4.14] on the transfer of coherence to
amalgamations. ThenextresultisacombinationofthislemmaandCorollary2.9
withthewell-knownfactthatRissemihereditaryifandonlyifRiscoherentand
w.dim(R)≤1[5,Theorem3.3].
Corollary 2.13. Suppose that f−1(J) and J are finitely generated in A and f(A)+J,
respectively. Then: A⊲⊳f J is semihereditary ifand only ifA is semihereditary, f(A)+J
is coherent arithmetical (resp., Coherent Gaussian), J∩Nil(B)=0 and, for every m∈
Max(A,f−1(J)),JSm =0or fm−1 JSm =0.
(cid:0) (cid:1)
SeeExample3.3whereA⊲⊳f Jissemihereditaryand f(A)+Jisnotsemiheredi-
tary. Corollary2.13recoversaknownresultforduplications:
Corollary2.14([12,Theorem4.1(2)]). SupposeIisfinitelygenerated. Then:
A⊲⊳IissemihereditaryifandonlyifAissemihereditaryandIm=0, ∀ m∈Max(A,I).
Contrastthisresultwith[19,Corollary4.15]onthetransferofthesemihereditary
propertytoamalgamations.
3. Examples
First, as an illustrative example for Theorem 2.1, we provide a family of non-
reducedarithmeticalringswhichariseasbi-amalgamations.
Example3.1. Let(A,m) be a valuationdomain, K:=qf(A), E a finitely generated
A−modulesuchthatEm=0,andB:=A⋉EthetrivialringextensionofAbyE,and
C:=K[[X]]. Consider the natural injective ring homomorphisms f :A֒→B and
g:A֒→C and let J:=0⋉E. We claim that the bi-amalgamation R:=A⊲⊳f,g (J,0)
is a non-reduced arithmetical ring. Indeed, notice first that f−1(J)= g−1(0)=0,
f(A)+J = B, and g(A)= A. Further, B is (local) arithmetical by [28, Theorem
3.1]. So, R is a chained ring by Theorem 2.1. However, R is not reduced by [26,
Proposition4.7]asJ2=0.
Next, as an illustrative example for Corollary 2.4, we provide a non-reduced
arithmeticalringwhicharisesasanamalgamation.
Example 3.2. Let (A,m) be a valuation domain, E a nonzero divisible A-module
whose submodules are totally ordered by inclusion (e.g., E:=qf(A)), and B:=
A⋉E the trivial ring extension of A by E. Consider the natural injective ring
homomorphism f :A֒→Bandlet J:=0⋉E. Then, theamalgamationR:=A⊲⊳f J
BI-AMALGAMATIONSSUBJECTTOTHEARITHMETICALPROPERTY 9
isanon-reducedarithmeticalring. Indeed, f−1(J)=0and f(A)+J=Bisachained
ringby[3,Theorem4.16]. So,RisachainedringbyCorollary2.4butnotreduced
by[15,Proposition5.4]since J∩Nil(B),0.
The converses of Corollary 2.8 and Corollary 2.11 are not true in general. A
counter-example is given below in the trivial case of an amalgamation where
A⊲⊳f J(cid:27)Aand f(A)+J=B.
Example3.3. Considerthecanonicalsurjectiveringhomomorphism f :Z→Z/4Z
andlet J denotethezeroidealofZ/4Z. Then,Z⊲⊳f J(cid:27)ZisaDedekinddomain
and f(Z)+J=Z/4Zisnotreducedsothatw.dim(Z/4Z)>1.
Next, in order to illustrate Corollary 2.8, one may use bi-amalgamations to
enrich the literature with new examples of non-semihereditary rings with weak
globaldimension≤1fromtheexistingones.
Example 3.4. Let A be a Noetherian ring with w.dim(A ) ≤ 1 and let I be a
o o
proper ideal of Ao such that Im =0, ∀ m∈Max(Ao,I) e.g.,Ao :=Z/12Z and I:=
4Z/12Z; clearly, Im1 =0 and Im2 =0, where m1 :=2Z(cid:16)/12Z and m2 :=3Z/12Z .
LetA:=A ⊲⊳I bethe amalgamatedduplicationof A alongI. ByCorollary2.10(cid:17),
o o
w.dim(A)≤1. Let D be any non-coherent ring with w.dim(D)≤1 (e.g., [24, Ex-
ample 4.1]). Finally, consider the natural ring homomorphisms f :A։A and
o
g: A ֒→A×D, and let J :=I and J′ :=(I ⊲⊳I)×D. Then, the bi-amalgamation
R:=A⊲⊳f,g (J,J′) is a non-semihereditary ring with weak global dimension ≤1.
Indeed, notice first that f−1(J)=g−1(J′)=I⊲⊳I. Then, we have w.dim(f(A)+J)=
w.dim(A )≤1 and w.dim(g(A)+J′)=w.dim((A×0)+((I⊲⊳I)×D))=w.dim(A×
o
D)=sup{w.dim(A),w.dim(D)}≤1. Moreover, let m⊲⊳I∈Max(A,I⊲⊳I). Neces-
sarily,m∈Max(Ao,I). Therefore,Sm⊲⊳I = f(A−(m⊲⊳I))+J=(Ao−m)+I andhence
JSm⊲⊳I =Im =0. Now, Ao and D are reduced and so are A and A×D. By Corol-
lary 2.8, w.dim(R)≤1. Finally, note that R is not coherent (and, a fortiori, not
semihereditary)since R (cid:27)g(A)+J′=A×Disnotcoherent(asDisnotcoherent).
J×0
Next,asanillustrativeexampleforCorollary2.11,weprovideanewexample
ofsemihereditaryringwhicharisesasabi-amalgamation.
Example3.5. LetAbeaNoetheriansemihereditaryringandletIbeaproperideal
of A such that Im =0, ∀ m∈Max(A,I) e.g., A:=Z/12Z and I:=4Z/12Z . Let
B:=A⊲⊳Ibetheamalgamatedduplicati(cid:16)onofAalongIandletDbeaNoeth(cid:17)erian
semihereditaryring. Finally, consider the naturalinjective ring homomorphisms
f :A֒→Band g:A֒→A×D,andlet J:=I⊲⊳I=I×I and J′:=I×D. Then, thebi-
amalgamationR:=A⊲⊳f,g(J,J′)isaNoetheriansemihereditaryring. Indeed,note
that f−1(J)=g−1(J′)=Iand,∀ m∈Max(A,I),Sm:= f(A−m)+I×I= (A−m)+I ×
(cid:16) (cid:17)
10 S.KABBAJ,N.MAHDOU,ANDM.A.S.MOUTUI
(A−m)+I . SothatJSm =(I×I)Sm(cid:27)Im×Im=0.Moreover, f(A)+J=A⊲⊳A=A×A
a(cid:16)nd g(A)+(cid:17)J′=A×0+I×D=A×DareNoetheriansemihereditaryrings(sinceso
areAandD). Therefore,Corollary2.11leadstotheconclusion.
References
[1] J.Abuihlail,M.JarrarandS.Kabbaj,Commutativeringsinwhicheveryfinitelygeneratedidealis
quasiprojective,J.PureAppl.Algebra215(2011)2504–2511.2
[2] K.AlaouiIsmailiandN.Mahdou,Coherenceinamalgamatedalgebraalonganideal,Bull.Iranian
Math.Soc.41(3)(2015)625–632.8
[3] D.D.AndersonandM.Winders,Idealizationofamodule,J.Commut.Algebra1(1)(2009)3–56.9
[4] C. Bakkari, S.KabbajandN.Mahdou, TrivialextensionsdefinedbyPru¨ferconditions, J.Pure
Appl.Algebra214(2010)53–60.2
[5] S. Bazzoni and S. Glaz, Pru¨fer rings, in: J. Brewer, S. Glaz, W. Heinzer, B. Olberding (Eds.),
Multiplicative IdealTheoryinCommutativeAlgebra: Atributetothework ofRobert Gilmer,
Springer,NewYork,2006,pp.55–72.2,7,8
[6] S.BazzoniandS.Glaz,Gaussianpropertiesoftotalringsofquotients,J.Algebra310(2007)180–193.
2
[7] M.BoisenJr.andP.Sheldon, CPI-extension: overringsofintegraldomainswithspecialprime
spectrum,Canad.J.Math.29(1977)722–737.1
[8] J.G.Boynton,Pullbacksofarithmeticalrings,Comm.Algebra35(2007)2671–2684.2,5
[9] J.G.Boynton,PullbacksofPru¨ferrings,J.Algebra320(2008)2559–2566.2,5
[10] J.G.Boynton,Pru¨ferconditionsandthetotalquotientring,Comm.Algebra39(5)(2011)1624–1630.
2
[11] H.Cartan,S.Eilenberg,HomologicalAlgebra,PrincetonUniversityPress,1956.2
[12] M.Chhiti,M.Jarrar,S.KabbajandN.Mahdou,Pru¨ferconditionsinanamalgamatedduplication
ofaringalonganideal,Comm.Algebra43(1)(2015)249–261.1,2,6,7,8
[13] F.Couchot,Gaussiantrivialringextensionsandfqp-rings,Comm.Algebra43(7)(2015)2863–2874.
2
[14] M.D’Anna,AconstructionofGorensteinrings,J.Algebra306(6)(2006)507–519.1
[15] M.D’Anna,C.FinocchiaroandM.Fontana,Amalgamatedalgebrasalonganideal,in:M.Fontana,
S.Kabbaj,B.Olberding,I.Swanson(Eds.),CommutativeAlgebraanditsApplications,Walterde
Gruyter,Berlin,2009,pp.155–172.1,9
[16] M.D’Anna,C.FinocchiaroandM.Fontana,Propertiesofchainsofprimeidealsinanamalgamated
algebraalonganideal,J.PureAppl.Algebra214(9)(2010)1633–1641.1,6
[17] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic
properties,J.AlgebraAppl.6(3)(2007)443–459.1
[18] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-
canonicalideal,Ark.Mat.45(2)(2007)241–252.1
[19] C.Finocchiaro,Pru¨fer-likeconditionsonanamalgamatedalgebraalonganideal,HoustonJ.Math.
40(1)(2014)63–79.1,2,3,6,7,8
[20] L.Fuchs,UberdieIdealearithmetischerRinge,Comment.Math.Helv.23(1949)334–341.2
[21] S.Glaz,Commutativecoherentrings,LectureNotesinMathematics,1371,Springer-Verlag,Berlin,
1989.
[22] S. Glaz, The weak global dimension of Gaussian rings, Proc. Amer. Math. Soc. 133 (9) (2005)
2507–2513.2
[23] S.Glaz,Pru¨ferconditionsinringswithzero-divisors,CRCPressSeriesofLecturesinPureAppl.
Math.241(2005)272–282.2
[24] S.GlazandR.Schwarz,Pru¨ferconditionsincommutativerings,Arab.J.Sci.Eng.36(2011)967–983.
2,9
[25] C.U.Jensen,Arithmeticalrings,ActaMath.Hungr.17(1966)115–123.2
[26] S.Kabbaj,K.Louartiti,andM.Tamekkante,Bi-amalgamatedalgebrasalongideals,J.Commut.
Algebra,toappear.harXiv:1407.7074i1,3,4,5,6,7,8
[27] H.MaimaniandS.Yassemi,Zero-divisorgraphsofamalgamatedduplicationofaringalongan
ideal,J.PureAppl.Algebra212(1)(2008)168–174.1
[28] A.Mimouni, M.Kabbour, andN.Mahdou,Trivialringextensionsdefinedbyarithmetical-like
properties,Comm.Algebra41(2013)4534–4548.8