On minimal prime submodules

Abstract

It is shown that if every prime left ideal minimal over an ideal I is finitely generated, then there are only finitely many prime left ideals minimal over I. This immediately generalizes Cohen’s theorem. We also extend the former result to multiplication modules over commutative rings.

Full text

ON MINIMAL PRIME SUBMODULES ∗
M. Behboodi and H. Koohy
Abstract
It is shown that if every prime ideal minimal over an ideal Iis
finitely generated, then there are only finitely many prime left ideals
minimal over I. This immedeately generalizes Cohen’s theorem. We
also extend the former result to multiplication modules over commu-
tative rings.
Introduction. Let Mbe a left R-module. Then a proper submodule Pof
Mis called prime if, for any r∈Rand m∈Mwith rRm ⊆P, we have
m∈Por rM ⊆P. Equivalently, Pis prime if, for any ideal Aof Rand
any submodule Nof Msuch that AN ⊆P, either AM ⊆Por N⊆P. The
notion of prime submodule was first introduced and systematically studied
by Dauns [3] and recently has received some attention. Several authors have
extended the theory of prime ideals of Rto prime submodules, see [10], [5],
[8], [9], [12]. For example, the classical result of Cohen’s is extended to prime
submodules over commutative rings, namely a finitely generated module is
Noetherian if and only if every prime submodule is finitely generated, see
[8], Theorem 6 and [5] and also any Noetherian module contains only finitely
many minimal prime submodules, see [10], Theorem 4.2. Recently, D.D.
Anderson [1] has elegantly generalized the well-known counterpart of the
latter result for commutative rings. He abandoned the Noetherianess and
showed that if all the prime ideals minimal over an ideal Iof Rare finitely
generated, then there are only finitely many prime ideals minimal over I. In
this note we give two generalizations of Anderson’s result. First we show that
∗1991 Mathematics Subject Classification: 13E05
1

in fact this result of Anderson is true in any associative ring (not necessarily
commutative) and also observe that this gives a nice generalization of Cohen’s
Theorem. We also extend Anderson’s Theorem to minimal prime submodules
in multiplication modules over commutative rings and observe that our result
and Theorem 3.7 in [4] immediately show that if Mis a multiplication module
and every minimal prime submodule of Mis finitely generated, then Mis
finitely generated. This shows in particular that the extension of Cohen’s
Theorem for a multiplication module Mis true without the assumption that
Mis finitely generated, see the main theorem of [5] and [8] Theorem 6.
All rings are associative with identity and all modules are unital left
modules. A proper left ideal Pof the ring Ris called prime if aRb 6⊆
Pwhenever a and b do not belong to P, or equivalently if Pis a prime
submodule of R. Clearly, maximal left ideals and prime ideals are some
examples of prime left ideals. By Zorn’s Lemma one can easily see that
each prime submodule of a module Mcontains a minimal prime submodule
of M, see [10] for more results on minimal prime submodules. If a ring
Ris commutative, then an R-module Mis called a multiplication module
provided for every submodule Nof Mthere exists an ideal Iof Rsuch that
N=IM , see [2], [11], [4] for an extensive study of these modules.
We begin with the following which shows that Anderson’s result is true
for any ring.
Theorem 1. Let Ibe a proper ideal in a ring R. If every prime left ideal
minimal over Iis finitely generated, then there are only finitely many prime
left ideals minimal over I.
Proof. We first claim that prime left ideals minimal over Iand prime ideals
minimal over Iare the same objects. It suffices to show that if Pis a
prime left ideal minimal over I, then Pis a two sided ideal. To see this, let
Q={a∈R:aR ⊆P}, then clearly Qis the largest ideal of Rwhich is
contained in Pand contains I. We also note that Qis a prime ideal, for if
not, then there are two ideals Aand Bproperly containing Qwith AB ⊆Q,
then AB ⊆Pimplies that either A⊆Por B⊆P, i.e., A⊆Qor B⊆Q,
which is impossible, thus P=Q. Now set
S={P1P2. . . Pn: each Piis a prime ideal minimal over I}.
If Icontains a finite product of prime ideals minimal over I, then we are
2

through. Thus we may assume that Ais not contained in Ifor each A∈S
and get a contradiction. Put
T={B:Bis an ideal of Rwith I⊆Band A6⊆ Bfor each A∈S}.
Clearly, each element of Sis finitely generated, for if P1=Pn
i=1 Rai,P2=
Pm
j=1 Rbjare two ideals, then P1P2is finitely generated and P1P2=PRaibj,
i= 1,2, . . . , n,j= 1,2, . . . , m. Now by Zorn’s Lemma, Thas a maximal
element Pwhich is clearly a prime ideal. Now I⊆Pimplies that Pcontains
a prime ideal Qminimal over I. Thus Q∈S, a contradiction.
In [6], Corollary 1, it is shown that if a ring Rhas acc on prime ideals
and has only finitely many prime ideals minimal over any ideal, then every
prime ideal is minimal over some finitely generated subideal, thus we have
the following generalization of Cohen’s Theorem.
Corollary 1. If a commutative (even duo) ring Rhas acc on prime ideals
and each prime ideal minimal over a proper subideal is finitely generated,
then Ris Noetherian.
Proof. If every ideal in Ris prime, then we are through. Hence, let Ibe
a nonprime ideal in R, then by our assumption each prime ideal Pminimal
over Iis finitely generated, for I⊂Pis a proper subideal of P. Now in view
of our Theorem 1 there are only finitely many prime ideals minimal over I
(note; if Iis prime, then this latter statement is evident). Thus, we may now
invoke Corollary 1 in [6], to see that each prime ideal Qof Ris minimal over
a finitely generated subideal A. Hence, if Ais a proper subideal of Q, then
by our assumption, Qis finitely generated and if A=Q, then trivially Qis
finitely generated and therefore Cohen’s Theorem completes the proof.
Next, we are going to prove Andreson’s result for multiplication modules.
First, we need the following easy lemmas.
Lemma 1. Let Pbe a prime submodule of an R-module Mand A1, A2, ..., An
be ideals of Rwith A1A2. . . AnM⊆Pthen AkM⊆Pfor some 1 ≤k≤n.
Proof. Evident.
Lemma 2. Let Rbe commutative ring and Mbe an R-module and Aand B
be two ideals of Rsuch that AM and BM are finitely generated submodules
of M, then ABM is also finitely generated R-module.
3

Proof. Let AM =Pn
i=1 Raimiand BM =Pk
j=1 Rbjm0
j, where mi, m0
j∈M
and ai∈A,bj∈B, for all i, j. We claim that
ABM =Pi,j Raibjmi=Pi,j Raibjm0
j,i= 1,· · · , n,j= 1,· · · , k.
Clearly, Pi,j Raibjmi⊆ABM . Now each element of ABM is a finite sum
of elements of the form abm, where m∈M,a∈A,b∈B. But we have
abm =a(Pk
j=1 cjbjm0
j), where cj∈R. Hence abm =Pk
j=1 cjbj(am0
j) =
Pk
j=1 cjbj(Pn
i=1 diaimi), where di∈R. This shows that abm ∈Pi,j Raibjmi.
Similarly, we have ABM =Pi,j Raibjm0
j.
Now we are ready to prove the second generalization of Anderson’s The-
orem.
Theorem 2. Let Rbe a commutative ring and Mbe a multiplication
module and Na proper submodule of Msuch that every prime submodule
of Mminimal over Nis finitely generated, then there are only finitely many
prime submodules of Mminimal over N.
Proof. Let
S={A1A2. . . AnM: each AiMis a prime submodule minimal over N}.
If Ncontains an element of S, say A1A2...AnM, then any prime submodule of
Mminimal over Ncontains some AiMby Lemma 1, i.e., {A1M,...,AnM}
is the set of all prime submodules minimal over N. Therefore we may assume
that Ncontains no element of Sand get a contradiction. Now consider the
set T={K:N⊆Kis a submodule of Mcontaining no element of S}.
In view of Lemma 2 each element of Sis finitely generated and therefore
we can use Zorn’s Lemma to show that Thas a maximal element, say P.
We claim that Pis a prime submodule. If not, then there exist a∈R,
m∈M\Pwith am ∈Pand aM 6⊆ P. Now there exists m0∈Mwith
am06∈ P. Hence Pis properly contained in P+Rm and in P+Ram0.
Now by the maximality of Pthere exist elements N1and N2in Ssuch that
N1=A1A2· · · AnM⊆P+Rm and N2=B1B2· · · BkM⊆P+Ram0. Put
A=A1A2...Anand B=B1B2...Bkand let x∈Aand y∈Band m1∈Mbe
arbitrary elements. Then we have xym1=x(ym1) = x(p1+ram0) = xp1+
xram0=xp1+ra(p2+tm) = xp1+rap2+rtam ∈P, where p1, p2∈P,r, t ∈R.
This shows that ABM =A1A2· · · AnB1B2· · · BkM⊆P, a contradiction.
Next, we recall that if Ris a commutative ring and Mis a multiplication
R-module with only finitely many minimal prime submodules, then Mis
4

finitely generated, see [4], Theorem 3.7. This result and our Theorem 2
immediately yield the following.
Corollary 2. If Ris a commutative ring and Mis a multiplication R-module
such that every minimal prime submodule of Mis finitely generated, then
Mis also finitely generated.
In [5], [8] Cohen’s Theorem is proved for finitely generated modules.The
above Corollary shows that for multiplication modules we have the following.
Corollary 3. If Ris a commutative ring and Mis a multiplication R-
module, then Mis Noetherian if and only if, every prime submodule of M
is finitely generated.
Acknowledgment. The authors wish to express their gratitude to professor
O. A. S. Karamzadeh for the valuable advice and encouragement given
during the preparation of this paper.
References
[1] D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math.
Soc. 122(1994), 13-14.
[2] A. Barnard, Multiplication modules, J. Algebra 71(1981), 174-178.
[3] J. Dauns, Prime modules, J. Reine Angew. Math. 298(1978), 156-181.
[4] Z. A. EL-Bast and P. F. Smith, Multiplication modules, Comm. Algebra
16(1988), 755-779.
[5] H. I. Karakas, On Noetherian modules, METU J. Pure and Appl. Sci,
5(2)(1972), 165-168.
[6] O. A. S. Karamzadeh, On the classical Kurll dimension of rings, Fund.
Math. 117(1983), 103-108.
[7] K. Koh, On one sided ideals of a prime type, Proc. Amer. Math. Soc.
28(1971), 321-329.
[8] C. P. Lu, Prime submodules of modules, Comm, Math. Univ. Sancti.
Pauli 33(1984), 61-69.
5

[9] R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a
module over a commutative ring, Comm. Algebra, 25(1)(1997), 79-103.
[10] R. L. McCasland and P. F. Smith, Prime submodules of Noetherian
modules, Rocky Mountain J. Math. 23(1993), 1041-1062.
[11] S. Singh and F. Mehdi, Multiplication modules, Canad. Math. Bull.
22(1979) , 93-98.
[12] Y. Tiras, A. Harmanci and P. F. Smith, A Characterization of prime
submodules, J. Algebra 212(1999), 743-752.
Department of Mathematics, Chamran University, Ahvaz, Iran.
E-mail addresses:
mahmood_behboodi@yahoo.com
hkoohy@yahoo.com
6