Vol. 19 (2018), No. 1, pp. 125–139 DOI: 10.18514/MMN.2018.1670
ON STRONGLY PRIME SUBMODULES
ABDULRASOOL AZIZI Received 13 May, 2015
Abstract. LetRbe a commutative ring with identity andManR-module. A proper submodule N ofM is called strongly prime [resp. strongly semiprime], if..NCRx/WM /yN [resp.
..NCRx/WM /xN] forx; y2M implies thatx2N ory2N [resp. x2N]. Strongly prime and strongly semiprime submodules are studied, in this paper.
2010Mathematics Subject Classification: 13A10; 13C99; 13E05; 13F05; 13F15
Keywords: I-maximal submodule, prime submodule, strongly prime submodule, strongly semiprime submodule
1. INTRODUCTION
Throughout this paper all rings are commutative with identity and all modules are unitary. Also we consider R to be a ring and M a unitaryR-module. If N is a submodule ofM;then.NWM /D ft2RjtMNg:
Recall from [12] that a proper submodule N of M is said to be strongly prime [resp.strongly semiprime], if..NCRx/WM /yN [resp...NCRx/WM /xN] forx; y2M implies thatx2N ory2N [resp.x2N].
If N is a strongly prime [resp. strongly semiprime] submodule of M and.N W M /DI;then we say that N is anI-strongly prime [resp. I-strongly semiprime]
submodule ofM:
A proper submoduleN ofM isprimeif the conditionra2N; r 2Randa2M implies that eithera2N orrM N:In this case, ifP D.N WM /;we say thatN is aP-prime submodule ofM;and it is easy to see thatP is a prime ideal ofR(see, for example, [2–8,10,11,13]).
In [11], another notion of strongly prime submodules was introduced. It is easy to see that for modules over commutative rings with identity, the strongly prime submodules introduced in [11] coincide with the prime submodules. However, the strongly prime submodules in this paper are special cases of prime submodules (see (1)(2)).
In [12, Proposition 1.1], it is shown that:
(1) Every strongly prime submodule is strongly semiprime and also prime.
c 2018 Miskolc University Press
(2) Every maximal submodule is strongly prime.
A characterization of strongly prime submodules is given in (1), and there we prove that the converse of (1) is also correct, see also (2). The converse of (2) is studied in(3),(1), and(6).
Prime submodules have been the subject of numerous publications in the past.
So the relations between prime submodules and strongly prime submodules will be helpful in connecting the previous papers on the prime notion for modules to this new notion of prime. Especially there are a few papers studying the form of the elements of the intersection of prime submodules (for example, [3,7,10,13]). This is a motivation for studying the equality of the intersection of prime submodules with the intersection of strongly prime submodules in this paper (see(7)and(3)).
The Generalized Principal Ideal Theorem (GPIT) states that ifRis a Noetherian ring andP is a minimal prime ideal over an idealI generated bynelements ofR;
thenht P n:The module version of GPIT related to prime submodules has been studied in [6] and slightly in [5]. In [12, Theorem 2.3], it is proved that the module version of GPIT related to strongly prime submodules is true for every Noetherian flat module. This shows that the behavior of height of strongly prime submodules is closer to prime ideals in comparing with prime submodules, and strongly prime submodules behave better than prime submodules for studying the dimension theory in modules.
In [12, Theorem 1.7], it is claimed that any intersection of strongly prime submod- ules is a strongly semiprime submodule. We show that this claim is incorrect (see(1) and(4)).
The existence of strongly prime submodules are studied in(1),(4),(3),(9). The results(2) and(5)are devoted to studying the existence of strongly semiprime sub- modules. In (3) and (3) we show that strongly prime submodules exist rarely in injective modules, although (9) shows that they are found frequently in projective modules. We apply the notion of strongly prime submodules and show that if there exists a non-zero module over a primary ring that is both projective and injective, then the ring has only one prime ideal (see(6)).
2. STRONGLY PRIME AND STRONGLY SEMIPRIME SUBMODULES
LetI be an ideal ofR:A submoduleNofM is calledI-maximal, if.NWM /DI and the submodule N is maximal with respect to this property, that is, if L is a submodule ofM containingN with.LWM /DI;thenLDN (see [2] and [10, p.
1810]).
One can easily see that ifN isI-maximal andI is a prime [resp. maximal] ideal, thenN is a prime [resp. maximal] submodule (see [10, Lemma 3.2]).
The following result is a generalization of [12, Proposition 1.1]. This result also shows that strongly prime submodules are exactlyI-maximal submodules, whereI is a prime ideal ofR:
Proposition 1. LetN be a submodule ofM:Then the following are equivalent:
(1) N is a strongly prime submodule ofM.
(2) N is a strongly semiprime submodule and also a prime submodule ofM:
(3) N is a strongly semiprime submodule ofM and.NWM /is a prime ideal.
(4) N is.N WM /-maximal and.N WM /is a prime ideal ofR:
Proof. ((a)H)(b)). See [12, Proposition 1.1].
((b)H)(c)). Note that for every prime submoduleN ofM;the ideal.N WM /is prime.
((c)H)(d)). To show thatN is.NWM /-maximal, suppose thatLis a submodule ofM containingN with.LWM /D.NWM /:Letxbe an arbitrary element ofL:As N .NCRx/L;we have.N WM /..NCRx/WM /.LWM /D.N WM /;
that is,..NCRx/WM /D.N WM /:Then..NCRx/WM /xD.N WM /xN;and sinceN is a strongly semiprime submodule,x2N: This shows thatLDN; whih completes the proof.
((d)H)(a)). Let P D.N WM /:SinceN isP-maximal andP is a prime ideal, by [10, Lemma 3.2], N is a prime submodule; so if ..N CRx/ WM /y N; for x2M andy2MnN;then..NCRx/WM /.N WM /and evidently.N WM / ..NCRx/WM /:Hence..NCRx/WM /D.NWM /DP and sinceNisP-maximal,
x2.NCRx/DN;and sox2N:
Example1. LetP be a maximal ideal ofRandF a freeRmodule withrank F >
1:
(1) F contains aP-strongly prime submodule.
(2) F contains aP-prime submodule that is not strongly prime.
Proof. LetF D ˚i2˛R;and assume thati02˛:
(1) We show thatNDP˚ ˚i2˛; i¤i0R
is aP-strongly prime submodule ofF:
ObviouslyPF ˚i2˛P N;that isP .NWF /;and asP is a maximal ideal ofR;
we have.N WF /DP:Now letLbe a submodule ofF containingN with.LWF /D .N WF /:Since.LWF /DP is a maximal ideal,Lis aP-prime submodule ofF:If N¤L;then letfxigi2˛2LnN;and soxi0…P:Obviouslyf.1 ıi0i/xigi2˛2N L;and thusxi0fıi0igi2˛D fxigi2˛ f.1 ıi0i/xigi2˛2L;and sinceLis aP-prime submodule ofM;we haveei0D fıi0igi2˛2L:Also evidently for eachj2˛; j¤i0; we have ej D fıj igi2˛2N L:ThereforeLDF;which is a contradiction. This shows thatN is aP-strongly prime submodule submodule ofF:
(2) By [4, Corollary 2.9(i)] if M is a flat module andPM ¤M;thenPM is a P-prime submodule ofM:So the submodulePF is aP-prime submodule ofF and note thatPF N and.N WF /DP;hencePF is not a strongly prime submodule
ofF:
Recall that an idealI ofRis semiprime ifI Dp
I :Strongly semiprime submod- ules of a module are characterized in the following.
Proposition 2. A submoduleN ofM is strongly semiprime if and only if.NWM / is a semiprime ideal, andN is.N WM /-maximal.
Proof. (H)). To show that .N WM / is a semiprime ideal, on the contrary, let r 2p
.N WM /n.N WM /: Suppose that 1 < n is the smallest positive integer with rn2.NWM /;and let´2rn 1MnN:ThenR´rn 1MrM;and so..NCR´/W M /..NCrM /WM /:Hence:
..NCR´/WM /´..NCrM /WM /rn 1M Drn 1..NCrM /WM /M rn 1.NCrM /N;
and asN is a strongly semiprime submodule,´2N;which is impossible.
To show thatN is.N WM /-maximal, suppose thatLis a submodule ofM con- taining N with .L WM /D.N WM /: Let x be an arbitrary element of L: Then ..NCRx/WM /..LCRx/WM /D.LWM /D.NWM /:Then..NCRx/WM /x .N WM /xN;an sox2N;becauseN is a strongly semiprime submodule. This shows thatLDN;and thusN is.N WM /-maximal.
((H) Suppose..NCRx/WM /xN;wherex2M:Letr1; r22..NCRx/WM /:
Then r1r2M r1.NCRx/NCRr1xNC..NCRx/WM /xN; and so r1r22.N WM /: Consequently..NCRx/WM /2.N WM /; and so..NCRx/W M /p
.N WM /D.N WM /..NCRx/WM /;which implies thatNCRxDN;
by our assumption. Thusx2N:
The following corollary shows that iffNigi2˛ is a family of distinct P-strongly prime submodules ofM withj˛j> 1; then\i2˛Ni is never a strongly semiprime submodule. This corollary rejects [12, Theorem 1.7], which claims that any intersec- tion of strongly prime submodules is a strongly semiprime submodule.
Corollary 1. LetfNigi2˛be a family ofP-strongly prime submodules ofM:Then
\i2˛Ni is a strongly semiprime submodule if and only ifNi DNj for alli; j 2˛:
Proof. (H)). LetBD \i2˛Ni:Note that.BWM /D \i2˛.NiWM /DP D.NjW M /for eachj2˛:SinceBis strongly semiprime, by(2)it is.BWM /-maximal and BNj with.BWM /D.Nj WM /;soNj DB for allj 2˛;which completes the proof.
((H). Is evident.
Corollary 2. For a ringR;the following are equivalent:
(1) Spec.R/is a chain.
(2) Every strongly semiprime submodule of anyR-moduleM is strongly prime.
Proof. ((1)H)(2)). Every two prime ideals ofRare comparable, sop
.N WM / is an intersection of a chain of prime ideals, thus it is a prime ideal. By(2),.NWM /D p.N WM /;thus it is a prime ideal. Now by(1),N is a strongly prime submodule.
((2)H)(1)). LetP andQ be two prime ideals ofR: Thenp
P\QDp P\ pQDP \Q; that is,P\Qis a semiprime ideal, and so by our assumption it is a prime ideal. Now ifP 6QandQ6P;then considerp2P nQandq2QnP:
Thuspq2P\Q;butp62P\Q;andq62P\Q:
Recall that a commutative ring is primary if its zero ideal is primary (see [1, Chapter 4]). The set of zero-divisors ofRisZ.R/D fr2Rj 90¤s2R; rsD0g: Also
N.R/D fr2Rj 9n2N; rnD0g:
A characterization of primary rings is given in the following lemma.
Lemma 1. The following are equivalent:
(1) Z.R/N.R/:
(2) Z.R/DN.R/;and it is a prime ideal.
(3) Ris a primary ring.
Proof. ((1)H)(2)). We know thatZ.R/is a union of prime ideals, andN.R/is the intersection of all prime ideals. LetZ.R/D [i2˛Pi;where eachPi is a prime ideal, and leti02˛:Then[i2˛PiDZ.R/N.R/Pi0 [i2˛Pi:
((2)H)(3) and..3/H)(1)) are obvious.
Proposition 3. LetRbe a primary ring, M a divisibleR-module andN a sub- module ofM:Then the following are equivalent:
(1) N is a maximal submodule.
(2) N is a strongly prime submodule.
(3) N is a strongly semiprime submodule.
(4) N is a maximal submodule and.N WM /is the unique prime ideal ofR:
Proof. ((1)H)(2)H)(3))) see [12, Proposition 1.1].
((4)H)(1)). Is obvious.
((3)H)(4)). Let N be a strongly semiprime submodule ofM andL a proper submodule ofM containingN:We show that.N WM /D.LWM /:
On the contrary, letr2.LWM //n.NWM /:By(2),.N WM /is a semiprime ideal, soZ.R/N.R/p
.NWM /D.N WM /Iand hencer 2RnZ.R/: Since M is divisible,M DrMLM;which is a contradiction. Thus.NWM /D.LWM /;and by(2),N is.NWM /-maximal, thereforeNDL;and soN is a maximal submodule ofM:This also implies that.N WM /is a maximal ideal.
Ift2.NWM /nZ.R/;then asM is divisible,M DtMN;which is impossible.
Hence.N WM /Z.R/N.R/P for each prime idealP ofR;and as.N WM / is maximal,.NWM /DP;the ringRhas a unique prime ideal.N WM /:
Note4. The following corollary shows that in a divisible module over an integral domain (particularly in a vector space), even the intersection of two distinct maximal (strongly prime) submodules is not strongly semiprime, because it is not a maximal submodule. The following result is also a generalization of [12, Proposition 1.3].
Corollary 3. LetR be an integral domain and N a submodule ofM: Then the following are equivalent:
(1) M is an injective module andN is a maximal submodule.
(2) M is a divisible module andN is a strongly prime submodule.
(3) M is a divisible module andN is a strongly semiprime submodule.
(4) N is a maximal submodule andRis a field.
Proof. ((1)H)(2)H)(3)) and ((4)H)(1)) are obvious.
((3)H)(4)). The proof is given by(3).
Example2. LetRbe the field of real numbers and considerM DR2:By(3), the strongly prime submodules of M are exactly the maximal submodules of M;and we know that the maximal submodules of M are exactly the lines passing through the origin. Hence although M is a Noetherian module over a Noetherian ring, it has infinitely many minimal strongly prime submodules. Note that according to [11, Theorem 4.2], a Noetherian module over a Noetherian ring has finitely many minimal prime submodules.
For an idealI ofR;a primary decompositionID \niD1Qi will be called coprime, if either nD1orQi’s are coprime; equivalently whenp
Qi’s are coprime. When this is the case, because of the Second Uniqueness Theorem (see [1, Theorem 4.10]), I has a unique minimal primary decomposition. For example, in any semilocal ring R;for eachn2N;the ideal.J.R//n has a coprime primary decomposition. Also evidently each ideal of an Atrinian ring or a Noetherian domain of dimension one has a coprime primary decomposition. Let S DKŒx1; x2; x3; ; where K is a field andx1; x2; x3; are independent indeterminates and letIDS.x1 c1/n1.x1
c2/n2.x1 c3/n3 .x1 cm/nm;wherec1; c2; c3; ; cm are distinct element ofK;
andn1; n2; n3; ; nm2N:Then the idealI ofS has a coprime primary decompos- ition. Indeed \miD1S.x1 ci/ni is a coprime primary decomposition of I:Thus in the non-Noetherian infinite dimensional ringRDS=I;the zero ideal has a coprime primary decomposition.
Theorem 1. Let the zero ideal ofRhave a coprime primary decomposition. IfM is a divisibleR-module, then every strongly prime submodule ofM is maximal.
Proof. If the zero ideal ofRis primary, then the proof is given by(3). Now sup- pose that0D \niD1Qi is a coprime primary decomposition of the zero ideal. Con- sider the natural ring homomorphism' WR!.R=Q1/.R=Q2/ .R=Qn/;
'.r/D.rCQ1; rCQ2; ; rCQn/:Since0D \niD1Qi andQi’s are coprime, by [1, Proposition 1.10],'is a ring isomorphism. Hence without loss of generality, we can assume thatRDR1R2 Rn;where eachRi is a primary ring. Put M1D.1; 0; ; 0/M; M2 D.0; 1; 0; ; 0/M; ; Mn D.0; 0; ; 0; 1/M: One can easily see thatM1is anR1-module with the multiplication
r1 .1; 0; ; 0/m
D.1; 0; ; 0/ .r1; 0; ; 0/m
for eachr12R1andm2M:Sim- ilarly eachMi is anRi-module andM ŠM1M2 MnasR-modules. So we
can assume thatM DM1M2 Mnis a divisibleR1R2 Rn-module, and eachMi is anRi-module. Evidently for eachr12R1;we have:
(1) r12Z.R1/if and only if.r1; 1; 1; ; 1/2Z.R/:
(2) r1M1DM1if and only if.r1; 1; 1; ; 1/M DM:
Therefore asM is a divisibleR-module,M1is a divisibleR1-module. Similarly Mi is a divisibleRi-module, for eachi > 1:
LetT be a submodule ofM:It is easy to see that:
(3) T DT1T2 Tn;where eachTi is a submodule ofMias anRi-module.
(4) .T WM /D.T1WM1/.T2WM2/ .TnWMn/:
LetN be a strongly prime submodule ofM:Then by (3),NDN1N2 Nn; where eachNi is a submodule ofMi as anRi-module. We show that there exists a 1j nsuch thatNj is a strongly prime submodule ofMj andNiDMi for each i¤j:
By [9, p. 6, Exercise 1.2], the prime ideals ofR are of the formR1R2 Rj 1Pj RjC1 Rn;wherePj is a prime ideal ofRj:Thus as by (4) and (1),.N WM /D.N1WM1/.N2WM2/ .NnWMn/is a prime ideal ofR;there exists a1jnsuch that.Nj WMj/is a prime ideal ofRj and.NiWMi/DRi for eachi¤j:HenceNiDMifor eachi¤j:Now we show thatNj is a strongly prime submodule ofMj:
LetLj be a submodule ofMj containingNj with.Nj WMj/D.Lj WMj/:Con- siderLDM1M2 Mj 1LjMjC1 Mn:By (4),.NWM /D.LWM /;
and sinceNis a strongly prime submodule ofM andNL;by(1), we haveNDL;
which implies thatNj DLj:Thus again by(1),Nj is a strongly prime submodule of Mj:Now asRj is a primary ring, by(3),Nj is a maximal submodule ofMj:Hence by (3),N DM1M2 Mj 1NjMjC1 Mnis a maximal submodule
ofM:
The set of all prime [resp. strongly prime] submodules of an R-module M is denoted bySpec.M /orSpecR.M /[resp. SSpec.M /or SSpecR.M /]. Evidently if M is finitely generated, then SSpecR.M /¤¿:
Proposition 5. Let the zero ideal ofR have a coprime primary decomposition andd i m R > 0:Then there does not exist any non-zero finitely generated divisible R-module. Particularly there does not exist any non-zero finitely generated injective R-module.
Proof. LetM be a non-zero finitely generated divisibleR-module. According to the proof of (1), we can assume that RDR1R2 Rn;where eachRi is a primary ring, andM DM1M2 Mnis anR1R2 Rn-module, where eachMiis a non-zero finitely generated divisibleRi-module, and so SSpecRi.Mi/¤
¿:Then(3)implies thatRi has a unique prime idealPi;and hence the prime ideals ofR are exactlyPbi DR1R2 Ri 1PiRiC1 Rn;for1i n:
EvidentlyPbi6cPj;for anyi¤j:This shows thatd i m RD0;which is a contradic-
tion.
LetSDRnZ.R/:ThenSis a multiplicatively closed subset ofR;and the ringRS
is called the total quotient ring ofR:SinceS contains no zero divisors, the natural map R!RS is injective, so RS is a ring extension ofR;and evidently RS is an R-module.
Lemma 2. LetN be a submodule ofM with.N WM /DP:
(1) N is a maximal submodule if and only ifN is a strongly prime submodule andP is a maximal ideal ofR:
(2) IfP is a maximal ideal, then there exists aP-strongly prime (maximal) sub- module ofM containingN:
Proof. (1) The proof is evident by(1).
(2) Note that M=N is a vector space over the field R=P, so it has a maximal subspace, say M=N:ThenMis a maximal submodule of M containingN;and as
P D.NWM /.MWM /;we haveP D.MWM /:
Corollary 4. Let the zero ideal ofRhave a coprime primary decomposition, and letKbe the total quotient ring ofR:Then the following are equivalent:
(1) d i m RD0:
(2) There exists a P-strong prime R-submodule of K;for each minimal prime idealsP ofR:
(3) KDR:
Proof. Evidently K is a divisible R-module. Let 0D \niD1Qi be a coprime primary decomposition of the zero ideal and p
Qi DPi: Then the minimal prime ideals ofRare exactlyP1; P2; ; Pn;alsoZ.R/D [niD1Pi:
((a)H)(b)). Asd i mRD0;the idealsP1; P2; ; Pnare all of the maximal ideals ofR:Obviously eachPi is a proper submodule ofKas anR-module. We show that .Pi WK/DPi;and so by(2)(2), there exists aPi-strongly primeR-submodule ofK;
which completes the proof.
Leta=s2K;wherea2Rands2RnZ.R/:Ass…Z.R/D [niD1Pi;it is a unit element of R; and hence for eachpi 2Pi; we have pi.a=s/D.s 1pia/=12Pi: Consequently PiK Pi;that is Pi .Pi W K/;and since Pi is a maximal ideal, Pi D.Pi WK/:
((b)H)(c)). By our assumption there exists aPi-strongly prime submoduleNi ofK as anR-module, and(1)implies thatNi is a maximal submodule ofK:Hence for each1in;the idealPi is a maximal idealR;and so the idealsP1; P2; ; Pn; are all of the maximal ideals ofR:Now letb=t2K;whereb2Randt2RnZ.R/:
Ast …Z.R/D [niD1Pi;it is a unit element ofR;and henceb=tD.t 1b/=12R:
ConsequentlyKDR:
((c) H) (a)). SinceK DR;the R-moduleK is a non-zero finitely generated
divisible module, and so by(5),d i m RD0:
Recall from [13] that anR-moduleM is said to bespecial, if for each maximal ideal M of R, every a2 M and any m2 M, there exist c 2RnM and k 2N such thatcakmD0. Semi-simple modules(direct sum of simple modules),locally Artinian modules(modules in which every cyclic submodule is Artinian) andsemi- Artinian modules(modules of which every homomorphic image has a nonzero simple submodule) are special (see [13, Section 3]).
Proposition 6.
(1) Every strongly prime submodule of a special module is a maximal submodule.
(2) d i m RD0if and only if every strongly prime submodule of anyR-module is maximal.
Proof. (1) LetN be a strongly prime submodule ofM andMa maximal ideal of Rcontaining.N WM /:Considerm2MnN;anda2M:SinceM is special, there existc2RnMandk2Nsuch thatcakmD0:Thencakm2N and asN is a prime submodule andc62.N WM /andm…N;we havea2.NWM /;and so.NWM /DM:
This shows that.NWM /is a maximal ideal ofR;and so by(2)(1),N is a maximal submodule ofM:
(2) Letd i m RD0:By [13, Theorem 3.5], every module over a zero dimensional ring is special, so the proof is given by part (1).
For the ’if part’ it is enough to considerRas anR-module.
Example3.
(1) Consider theZ-moduleM DZ=nZ;wherenis a positive integer. Evidently each semiprime ideal of Z is an ideal generated by a square free integer.
It is easy to see that strongly semiprime submodules ofM are of the form kZ=nZ;wherekis a square free integer withkjn:
(2) More generally let R be a Dedekind domain and consider the cyclic R- moduleM DR=I;whereI is a proper ideal of R: If the prime factoriza- tion ofI isP1˛1P2˛2 Pm˛m;then the strongly semiprime submodules ofM areP1ˇ1P2ˇ2 Pmˇm=I;where eachˇi is either 0 or 1, and at least for one 1im; ˇi D1:
(3) Each strongly prime submodule of theZ-moduleM DZ=nZis of the form pZ=nZ;where p is a prime factor of n; so they are exactly the maximal submodules ofM:
The following result guarantees the existence of anI-strongly semiprime submod- ule in a finitely generated moduleM;for any semiprime idealI containingAnn M:
Theorem 2. LetB be a submodule of a finitely generated R-moduleM: If.BW M /I, whereI is a proper semiprime ideal ofR, then there exists anI-strongly semiprime submodule ofM containingB:
Proof. LetT D fC M jBC; .C WM /Ig. By Zorn’s Lemma T has a maximal element, sayN:We prove thatN is anI-strongly semiprime submodule of M:
According to [9, Ex. 2.2., p. 13], ifM is finitely generated andI an ideal ofR;
thenp
.IM WM /Dp
Ann.M /CI :Hence:
..NCIM /WM /D.IM N W M
N/ r
.IM N W M
N/D r
Ann.M N /CI Dp
.N WM /CI DI:
SoNCIM 2T;and sinceN is a maximal element ofT;we haveN DNCIM. HenceI..NCIM /WM /D.NWM /I;and thus.N WM /DI.
Now suppose thatLis a submodule ofM containingN with.LWM /D.N WM /:
Then asNis a maximal element ofT;evidentlyLDN:Hence by(2),N is a strongly
semiprime submodule ofM:
Corollary 5. Let B be a submodule of a finitely generatedR-moduleM: Then there exists ap
.BWM /-strongly semiprime submodule ofM containingB:
The following result is a generalization of [5, Lemma 4] or [10, Theorem 3.3].
Lemma 3. Let B be a submodule of M andP a prime ideal of R: ThenB is contained in aP-strongly prime submodule, if one of the following holds:
(1) M is a finitely generatedR-module and.BWM /P:
(2) M is a specialR-module andBis aP-prime submodule ofM:
Proof. (1) By(2), there exists aP-strongly semiprime submoduleN ofM con- tainingB;and since.NWM /DP is a prime ideal,(1)implies thatN is aP-strongly prime submodule ofM:
(2) Follow the proof of(6)(1) to see that ifBis aP-prime submodule of a special module, thenP is a maximal ideal. Now by(2)(2),B is contained in aP-strongly
prime and containingB:
Lemma 4. [2, Theorem 2.8] If M is a finitely generated module, then everyP- prime submodule of M is an intersection of P-maximal submodules of M.
LetB be a proper submodule of anR-moduleM: The intersection of all prime submodules of M containing B is denoted by rad.B/ or radR.B/:If M has no prime submodules containingB;then we considerrad.B/DM:Also according to [12], the intersection of all strongly prime submodules ofM containingBis denoted by s-rad.B/or s-radR.B/:IfM has no strongly prime submodules containingB;
then we say s-rad.B/DM:
The following result shows that s-rad.B/coincides withrad.B/;for finitely gen- erated or special modules.
Proposition 7. s-rad.B/Drad.B/; for every submodule B of M; if M is a finitely generated or a special module.
Proof. By (3), rad.B/DM if and only if s-rad.B/DM: Now suppose that rad.B/¤M:As each strongly prime submodule is a prime submodule,rad.B/
s-rad.B/:
First suppose that M is a finitely generated module. For proving the converse inclusion, note that by(4), everyP-prime submodule ofM is an intersection of P- maximal submodules of M and by(1), the P-maximal submodules of M are exactly theP-strongly prime submodules ofM: Hence everyP-prime submodule ofM is an intersection of strongly prime submodules. This shows that s-rad.B/rad.B/;
whenM is finitely generated.
Now assume thatM is a special module. From the proof of(6), for anyP-prime submoduleN ofM;the idealP is maximal.
It is well-known that in a vector space the zero subspace is an intersection of maximal subspaces, so for the vector space M=N over the field R=P; we have N=N D \i2˛MNi;where each MNi is a maximal subspace of M=N: Then for each i 2˛; MNi DNi=N;whereNi is a maximal submodule ofM containingN:Hence N D \i2˛Ni;that is, every prime submodule of M is an intersection of strongly
prime submodules, and this completes the proof.
Example4. By(3), theZ-moduleQof all rational numbers has no strongly prime submodules, although the zero submodule is a prime submodule. Hence s-rad.0/D Qandrad.0/D0;that is, s-rad.0/¤rad.0/:This example and also the following result show that(7)does not hold in general.
Theorem 3. For a ringR;the following are equivalent:
(1) d i m RD0:
(2) s-rad.B/Drad.B/;for everyR-moduleM and anyBM:
(3) s-rad.0/Drad.0/;for everyR-moduleM:
Proof. ((i)H)(ii)) is given by(7), and ((ii)H)(iii)) is obvious.
((iii)H)(i)). LetP be a prime ideal ofR:We show thatP is a maximal ideal of R:First we prove that s-radR=P.0/DradR=P.0/;for everyR=P-moduleM0:
Evidently, we can considerM0as anR-module, by the natural pull back homo- morphismR!R=P;that is, we definerxD.rCP /x;for eachr2Randx2M0: LetN0be a prime submodule ofM0as anR=P-module. It is easy to check that:
(1) ()N0=N0is a prime submodule ofM0=N0as anR-module.
(2) ()If forN0N M0, N=N0is a strongly prime submodule ofM0=N0 as anR-module, thenN is a strongly prime submodule of M0 as anR=P- module.
By our assumption and(),N0=N0is an intersection of strongly semiprime sub- modules ofM0=N0as anR-module. Now by(),N0is an intersection of strongly
semiprime submodules of M0 as an R=P-module. Consequently s-radR=P.0/D radR=P.0/;for everyR=P-module module.
Now suppose thatKis the quotient field ofR=P:It is easy to see that for theR=P- moduleK;we haveSpecR=P.K/D f0g;and so s-radR=P.0/DradR=P.0/D0for
theR=P-moduleK:Thus by(4),R=P is a field.
Let S be a multiplicatively closed subset of R: For anyN MS; we consider NcD fx2M jx=12Ng:
Lemma 5. [8, Proposition 1] Let M be an R-module and S a multiplicatively closed subset ofR:
(1) IfN is aP-prime submodule ofM withP\SD¿;thenNS is aPS-prime submodule ofMS as anRS-module and.NS/cDN:
(2) IfT is aQ-prime submodule ofMS as anRS-module, thenTc is aQc- prime submodule ofM; .Tc/SDT andQc\SD¿:
The following result is a generalization of [12, Theorem 1.5 and Corollary 1.6].
Proposition 8. LetM be anR-module,S a multiplicatively closed subset ofR;
andN aP-strongly prime submodule ofM withP\SD¿:
(1) NS is aPS-strongly prime submodule ofMS as anRS-module and.NS/cD N:
(2) IfP is a maximal element of fI jI is an ideal ofRwithI\SD¿g;then NS is a maximal submodule ofMS:
(3) IfM is a finitely generated module, thenNSis a maximal submodule ofMSif and only ifP is a maximal element of the setfI jI is an ideal ofRwithI\ SD¿g:
(4) LetT be aQ-strongly prime submodule ofMS as anRS-module, whereM is finitely generated or a special module, orQcis a maximal ideal ofR:Then Tc is aQc-strongly prime submodule ofM:
Proof. (i) AsNis a strongly prime submodule, it is a prime submodule, and so ac- cording to(5)(i),NS¤MS and.NS/cDN:ThenNS is a strongly prime submodule ofMS;by [12, Theorem 1.5].
(ii) LetMbe a maximal ideal ofRS containingPS:ThenP D.PS/cMc and Mc\SD¿:Now sincePis a maximal element of the setfIjI is an ideal ofRwithI\ SD¿g;we haveP DMc;and soPS D.Mc/S DM:ThusPS is a maximal ideal ofRS;and by part (i),NS is aPS-strongly prime submodule, so by(2)(1),NS is a maximal submodule ofMS:
(iii) LetNS be a maximal submodule ofMS;and suppose AD fI P jI is an ideal ofRwithI\SD¿g:
By Zorn’s lemmaAhas a maximal element, sayM0: By(3), N is contained in an M0-strongly prime submoduleN0ofM:Part (i) implies thatNS0 is a strongly prime
submodule ofMS containingNS;and sinceNS is a maximal submodule,NS DNS0; and so N D.NS/cD.NS0/c DN0:ThusP D.N WM /D.N0WM /DM0;which implies thatP is a maximal element offIjI is an ideal ofRwithI\SD¿g:
(iv) SinceT is aQ-prime submodule,(5),(ii), implies thatTcis aQc-prime sub- module ofM:
IfM is finitely generated or a special module, then according to(3), Tc is con- tained in a Qc-strongly prime submoduleLofM: ThenT D.Tc/S LS and by (5)(i),.LS WMS/DQ:According to(1), T isQ-maximal, henceT DLS and by (5)(i),TcD.LS/cDL:ThenTc is aQc-strongly prime submodule ofM:
Now letQc be a maximal ideal ofR:IfTc is contained in a submoduleU ofM with.Tc WM /D.U WM /;then asQcD.U WM /is a maximal ideal,U is a prime submodule and(5)(i) implies that.US WMS/DQ:EvidentlyT D.Tc/S US and Q D.T WMS/D.US W MS/; and as T is a strongly prime submodule, we have T DUS;which implies that Tc DU:This shows thatTc is a Qc-strongly prime
submodule ofM:
The following example shows that the contraction of a strongly prime submodule is not necessarily strongly prime, in general.
Example 5. Let P be the zero ideal of Z: Then ZP DQ and QP DQ: So SSpecZP.QP/D f0g; although 0c D0…SSpecZ.Q/; because SSpecZ.Q/D¿; by(3).
From (3) we can conclude that injective modules over a primary ring with more than one prime ideal or particularly over an integral domain which is not field, have no strongly prime submodules. The following result shows the opposite of this for projective modules, the dual of injective modules.
Proposition 9. LetM be a non-zero a projective module, andP a prime ideal of R:ThenM has a strongly prime submodule containingPM if and only ifPM¤M:
Proof. LetPM ¤M:By [4, Corollary 2.9(i)] ifM is a flat module andPM ¤M;
then PM is a P-prime submodule of M: Now by (5)(i), .PM /P is a PP-prime submodule ofMP:Particularly this shows thatMP ¤0:
Now suppose thatMis a maximal ideal ofRcontainingP:By.MM/PM ŠMP andMP ¤0;we haveMM¤0:We show thatMMhas a maximal submodule. Since MM is a non-zero projective module over the local ringRM;it is a non-zero free RM-module. Hence we can assume thatMMis a direct sum of copies ofRM:
IfMMDRM;then obviouslyMMhas a maximal submodule (ideal in this case).
Now let MMD ˚i2˛RM; where j˛j> 1; and assume that i0 2˛; Consider T D MM˚ ˚i2˛; i¤i0RM
:ObviouslyMMMM ˚i2˛MMT;that isMM.T W MM/;and asMMis a maximal ideal ofRM;we have.T WMM/DMM: Thus by (2)(2), T is contained in an MM-strongly prime submodule Mof MM: Therefore as .MM/c DM is a maximal ideal of R;by (8)(iv), Mc is an M-strongly prime submodule ofM containingPM:
The ’only if’ part is obvious.
Recall that a ringRis semi simple, if and only if allR-modules are projective, if and only if allR-modules are injective (see [14, Proposition 4.5]).
Corollary 6. Let the zero ideal of R have a coprime primary decomposition. If there exists a non-zero R-module that is both projective and injective, then R has finitely many prime ideals andd i m RD0:
Proof. LetM be a non-zeroR-module that is both projective and injective. By the proof of (1), we can assume that RDR1R2 Rn;where eachRi is a primary ring, andM DM1M2 Mn;where eachMi is a non-zero divisible Ri-module. We show thatM1is a projectiveR1-module.
Letf1WA1!B1be a surjectiveR1-module homomorphism, and letg1WM1! B1 be an R1-module homomorphism. For each 2 i n; consider Ai DBi D 0; and let ADA1A2 An to B DB1B2 Bn: Define f W A! B; f .a1; a2; a3; ; an/D.f1.a1/; 0; 0; ; 0/andgWM !B; g.a1; a2; ; an/D .g1.a1/; 0; 0; ; 0/:Obviouslyf is a surjectiveR-module homomorphism andgis anR-module homomorphism.
As M is a projective R-module, there exists an R-module homomorphismhW M !Awith f ıhDg:Consider 1WA!A1; 1.a1; a2; ; an/Da1;and`1W M1!M; `1.a1/D.a1; 0; ; 0/;and leth1WM1!A1; h1D1ıhı`1:One can easily see that h1 is anR1-module homomorphism withf1ıh1Dg1: This shows thatM1 is a projectiveR1-modules. Similarly Mi is a projective Ri-modules, for each2in:
As Mi ¤0 for each1i n; we have MPi ¤0;for some prime idealPi of Ri;and by the proof of(9), SSpecRi.Mi/¤¿:Now as eachMi is a divisibleRi- module, by the proof of(5),d i m RD0and eachRi has a unique prime ideal, and
thereforeRhas finitely many prime ideals.
REFERENCES
[1] M. F. Atiyah and I. Macdonald,Introduction to commutative algebra. Student economy edition., student economy edition ed. Boulder: Westview Press, 2016.
[2] A. Azizi, “Intersection of prime submodules and dimension of modules.”Acta Math. Sci., Ser. B, Engl. Ed., vol. 25, no. 3, pp. 385–394, 2005.
[3] A. Azizi, “Radical formula and prime submodules.”J. Algebra, vol. 307, no. 1, pp. 454–460, 2007, doi:10.1016/j.jalgebra.2006.07.006.
[4] A. Azizi, “Height of prime and weakly prime submodules.”Mediterr. J. Math., vol. 8, no. 2, pp.
257–269, 2011, doi:10.1007/s00009-010-0068-6.
[5] A. Azizi and H. Sharif, “On prime submodules.”Honam Math. J., vol. 21, no. 1, pp. 1–12, 1999.
[6] S. M. George, R. L. McCasland, and P. F. Smith, “A principal ideal theorem analogue for mod- ules over commutative rings.” Commun. Algebra, vol. 22, no. 6, pp. 2083–2099, 1994, doi:
10.1080/00927879408824957.
[7] J. Jenkins and P. F. Smith, “On the prime radical of a module over a commutative ring.”Commun.
Algebra, vol. 20, no. 12, pp. 3593–3602, 1992, doi:10.1080/00927879208824530.
[8] C.-P. Lu, “Spectra of modules.”Commun. Algebra, vol. 23, no. 10, pp. 3741–3752, 1995, doi:
10.1080/00927879508825430.
[9] H. Matsumura,Commutative ring theory. Transl. from the Japanese by M. Reid. Paperback ed., paperback ed. ed. Cambridge: Cambridge University Press, 1989.
[10] R. McCasland and M. Moore, “Prime submodules.”Commun. Algebra, vol. 20, no. 6, pp. 1803–
1817, 1992, doi:10.1080/00927879208824432.
[11] R. McCasland and P. Smith, “Prime submodules of Noetherian modules.”Rocky Mt. J. Math., vol. 23, no. 3, pp. 1041–1062, 1993, doi:10.1216/rmjm/1181072540.
[12] A. Naghipour, “Strongly prime submodules.”Commun. Algebra, vol. 37, no. 7, pp. 2193–2199, 2009, doi:10.1080/00927870802467239.
[13] D. Pusat-Yilmaz and P. Smith, “Modules which satisfy the radical formula.”Acta Math. Hung., vol. 95, no. 1-2, pp. 155–167, 2002, doi:10.1023/A:1015624503160.
[14] J. J. Rotman,An introduction to homological algebra. 2nd ed., 2nd ed. Berlin: Springer, 2009.
doi:10.1007/b98977.
Author’s address
Abdulrasool Azizi
Shiraz University, Department of Mathematics, College of Sciences, Shiraz 71457-44776, Iran E-mail address:aazizi@shirazu.ac.ir a azizi@yahoo.com
