A.J. TAHERIZADEH
Communicated by the former editorial board
The concept of content modules and (weak) content algebras was introduced and developed in [6] and [7]. In [4] some new results for content algebra were proved.
In this article we obtain some results concerning (faithfully at) weak content algebras similar to those proved in [4] for content algebras. Also, the transfer of contentness of modules and algebras to trivial ring extension is considered.
AMS 2010 Subject Classication: 13C05, 13C99.
Key words: Content modules, content algebras, McCoy's property.
1. INTRODUCTION
Throughout this paper, all rings are commutative with non-zero identity and all modules are unitary. In this article, we discuss some properties of a special algebra called weak content algebra introduced in [7]. To do this, we need to recall the concepts of content modules and content algebras introduced in [6].
Let R be a commutative ring, and M an R-module. Then, the content function c, fromM to the set of all ideals of R is dened by (for allx∈M)
c(x) =∩{I :I is an ideal ofR and x∈IM}.
M is called a content R-module if x ∈ c(x)M, for all x∈ M. The concept of content modules stems from content function for polynomial rings and besides it is an interesting subject in the eld of module theory. For example, free modules and projective modules are content modules (see [6, p. 49), moreover Nakayama Lemma holds for content modules [4, Theorem 2].
Next, let R be a commutative ring and B an R-algebra. B is dened to be a content R-algebra, if the following hold.
1) B is a content R-module.
2) (faithful atness) For anyr∈Randb∈B,c(rb) =rc(b)andc(B) =R (by c(B) we mean the ideal ofR generated by all c(b), b∈B).
3) (Dedekind-Mertens content formula.) For each f and g in B, there exists a positive integer nsuch thatc(f)nc(g) =c(f)n−1c(f g).
MATH. REPORTS 15(65), 2 (2013), 153159
Examples of a content R-algebra are the group rings R[G] where G is a torsion-free abelian group [5]. For some examples of content R-algebra we refer the readers to [6, Examples 6.3] Rush dened weak content R-algebras as follows [7, p. 334]. An R-algebra R0 is called a weak content R-algebra if the following conditions hold:
1) R0 is a content R-module.
2) (weak content formula) for all f, g∈R0,c(f)c(g)⊆p c(f g).
It is obvious that content algebras are weak content algebras, but the converse is not true. For example, if R is a Noetherian ring, then R[[X1, X2, . . . , Xn]] is a (faithfully at) weak content R-algebra while it is not a content R-algebra [7, p. 331].
Our main purpose in this paper is to obtain some results about (faith- fully at) weak content algebra similar to those proved for content algebras in literature and we do this in Section 2.
In Section 3, we consider the transfer of contentness of modules and alge- bras to trivial extension of a ringR by a contentR-module or R-algebra.
2. WEAK CONTENT ALGEBRAS
Denition 2.1. Let M be an R-module and the content function e from M to the ideals of R is dened by
c(x) =∩{I : I is an ideal of R and x∈IM}.
M is called a contentR-module if x∈c(x)M, for all x∈M. Also when N is a non-empty subset of M, then by c(N) we mean the ideal of R generated by all c(x), x∈N.
Lemma 2.2 ([6, 1.2]). Let M be an R-module. Then the following state- ments are equivalent:
I) M is a content R-module,
II) For any non-empty family of ideals {Ii} of R (∩Ii)M =∩(IiM). Moreover, whenM is a contentR-module,c(x)is a nitely generated ideal of R, for all x∈M.
For the next corollary, we recall that an R-module M is said to be a multiplication module if for every submodule N ofM,N =IM for some ideal I of R.
Corollary 2.3. Let M be a faithful multiplication R-module. Then M is a content R-module.
Proof. The result follows from Lemma 2.2. (II) and the well-known result of multiplication modules that
∩(IiM) = (∩(Ii+ AnnM))M
for any non-empty collection {Ii} of ideals of R (see for example [1, Corollary 1.7]).
Remark 2.4. It is worth to mention that in general the converse of corol- lary 2.3 is not true; for example, ifF is a freeR-module of rank at least 2, then F is a (faithful) contentR-module (by [6, p. 49]) where it is not a multiplication module.
Let us recall the following proposition of [6] which will be used in sequel.
Proposition 2.5 (See [6, Corollary 1.6]). Let M be a contentR-module.
Then, the following are equivalent:
1) M is a at R-module.
2) For every r∈R and x∈M,rc(x) =c(rx).
Moreover, M is faithfully at if and only if M is at and c(M) =R. For the next lemma, we recall two facts concerning content modules and algebras.
Fact I. LetM be a contentR-module andx∈M. Thenx= 0 if and only if c(x) = 0.
Fact II. Let B be a faithfully atR-algebra which is a content R-module.
Thenc(1B) =Randc(ρ(r)) =Rrwherer∈Randρ:R→B is the structural homomorphism.
Lemma 2.6. Suppose that B is a weak content R-algebra such that R is reduced and g∈Z(B). Then there exists r ∈R\{0} such that rc(g) = 0. If B is a faithfully at R-algebra which is a content R-module, then the converse of the above statement is true.
Proof. Let g ∈ Z(B). Then there exists f ∈ B\{0} such that f g = 0.
Thus, c(f g) = 0and by the weak content formulac(f)c(g)⊆p
c(f g) =√ 0 = 0, and so c(f)c(g) = 0. The result follows.
Now suppose thatB is a faithfully atR-algebra such that it is a content R-module and r ∈ R\{0}, g ∈ B be such that rc(g) = 0. Then 0 =rc(g) = c(rg) and so, 0 =rg=ρ(r)g. We note that ρ(r) 6= 0, because a faithfully at morphism is injective.
Corollary 2.7. (McCoy's Property) Let B be a faithfully at weak con- tentR-algebra such thatR is reduced. Theng∈Z(B)if and only if there exists r ∈R\{0} such that rc(g) = 0.
For the next proposition we recall the following theorem from [7].
Theorem 2.8 (See [7, Theorem 1.2]). LetB be an R-algebra such that B is a content R-module. Then, the following are equivalent:
1) B is a weak content R-algebra.
2) For each prime idealpofR, eitherpB is a prime ideal ofB orpB =B.
Remark 2.9. It follows from 2.8 that if B is a faithfully at weak content R-algebra, then for each primep ofR,pB is a prime ideal ofB.
In the next proposition byAssR(M), we mean the set of associated prime ideals of R-moduleM.
Proposition 2.10. Let B be a faithfully at weak contentR-algebra and 06=M be anR-module. Thenp∈AssR(M)if and only ifpB ∈AssB(M⊗RB). Proof. Easily follows from 2.8 and the remark just before proposition.
Denition 2.11. A ring R has very few zero-divisors if Z(R) is a nite union of prime ideals of Ass(R). (See [4, Denition 8]).
In the next three theorems, (2.12, 2.14 and 2.15) R is a reduced ring.
Theorem 2.12. Suppose thatB is a faithfully at weak contentR-algebra.
ThenR has very few zero-divisors if and only if B has very few zero-divisors.
Proof. (→) The proof is similar to [4, Theorem 9], and we omit it.
(←)Suppose thatBhas very few zero-divisors andZ(B) =St
i=1qi where qi ∈Ass(B), for all1≤i≤t. We show thatZ(R) =St
i=1pi wherepi ∈AssR, for all 1≤i≤n. Suppose thatqi := (0 :B bi) for some 06=bi ∈B. Then it is easy to see that pi := qi∩R = (0 :Rc(bi)). Therefore,pi = (0 :Rri) for some generator ri of c(bi) and so pi ∈AssRR, for all 1≤i≤r. Next, let r ∈Z(R) and r0 ∈ R\{0} be such that rr0 = 0. Then ρ(rr0) = 0 where ρ : R → B is the structural homomorphism. But ρ is injection and so ρ(r0) 6= 0. Hence, ρ(r) ∈ Z(B) and ρ(r) ∈ qj, for some 1 ≤ j ≤ t and r ∈ pj. Next, suppose that r ∈ St
i=1pi. The r ∈ pj for some 1 ≤ j ≤ t and so rrj = 0. Therefore r ∈Z(R).
Now, we give the following denition from [3] and prove the two next results.
Denition 2.13. A ringRhas property (A), if each nitely generated ideal I ⊆Z(R)have a non-zero annihilator.
The class of commutative rings with property (A) is quite large. For example, rings with very few zero-divisors (for example, Noetherian rings) have property (A). For more examples concerning rings with property (A) we refer to [2, p. 2].
Theorem 2.14. Let R be an (A)-ring and B be a faithfully at weak content R-algebra. Then Z(B) is a nite union of prime ideals in Min(B) if and only if Z(R) is a nite union of prime ideals in Min(R). (Here Min(R) means the set of all minimal prime ideals of R.)
Proof. First of all, we note that by [3, Corollary 7] the map Min(R) → Min(B) given byp→pB is a bijection.
(→): Let Z(R) = St
i=1pi where pi ∈ Min(R), for all 1 ≤ i ≤ t. We shall show that Z(B) = St
i=1piB. Let b ∈Z(B). Then by Lemma 2.6 there exists r ∈R\{0} such that rc(b) = 0, so c(b) ⊆Z(R) and c(b) ⊆pi, for some 1≤i≤t. Thus,b∈c(b)B ⊆piB. Conversely, letb∈St
i=1piBand sob∈pjB, for some 1≤j≤t. Therefore, c(b)⊆pj ⊆Z(R). ButRhas property (A) and so there is r ∈R\{0} such that rc(b) = 0 or c(rb) = 0. Hence, ρ(r)b=rb= 0 withρ(r)6= 0. This means thatb∈Z(B).
Next, let Z(B) = Sn
i=1piB where piB ∈ Min(B), for all 1 ≤ i ≤ n. We shall prove that Z(R) = Sn
i=1pi. To see this, let r ∈ Z(R). Then there exists r0 ∈ R\{0} such that rr0 = 0 and so ρ(rr0) = 0. Hence, ρ(r)ρ(r0) = 0 and ρ(r0) 6= 0, that is ρ(r) ∈ Z(B) and ρ(r) ∈ piB, for some 1≤i≤n. Hence, r∈piB∩R=pi (by [4, Corollary 7]). Conversely, suppose that r ∈ pj, for some 1 ≤ j ≤ n. Then ρ(r) ∈ piB and so ρ(r) ∈ Z(B). Thus, by Lemma 2.6 there exists r0 ∈R\{0} such that 0 =r0c(ρ(r)) =r0.rR. Therefore,r ∈Z(R).
Theorem 2.15. Suppose that B is a faithfully at R-algebra which is a weak content R-algebra. Also, suppose that for each nitely generated ideal I of R there existsb∈B such that c(b) =I. Then the following are equivalent:
1) R is an (A)-ring.
2) For all b∈B, b6∈Z(B) if and only if c(b) is a regular ideal of R. Proof. (1) → (2): Suppose R has property (A) and b 6∈ Z(B). Then by Lemma 2.6, for all r ∈ R\{0}, rc(b) 6= 0 and so (0 :R c(b)) = 0. Hence, c(b) 6⊆ Z(R) and c(b) is a regular ideal of R. Next, suppose that c(b) is a regular ideal of R, i.e., c(b) 6⊆Z(R). Therefore, 0 :c(b) = (0), that is, for all r ∈R\{0},rc(b)6= 0. Thus,rb6= 0 for allr ∈R\{0}.
(2) → (1): Let I be a nitely generated ideal of R such that I ⊆Z(R). By our assumption, there exists b ∈ B such that c(b) = I. But c(b) is not a regular ideal of R and so by hypothesis b ∈ Z(B). Therefore, there exists r ∈ R\{0} such that rb = 0. Thus, 0 = c(rb) = rc(b) and hence, I has a non-zero annihilator.
We recall that an R-module M is said to have very few zero-divisors if ZR(M) is a nite union of prime ideal inAssRM.
Proposition 2.16. Let M be a at content R-module. Then AssRM ⊆ AssR. In addition, if M has very few zero-divisors then ZR(M) ⊆ ZR(R). Finally, if B is a faithfully at R-algebra and a content R-module then AssR(B) = AssR.
Proof. By [7, Lemma 3.1] for each 0 6= m ∈ M,AnnRm = AnnRc(m). Now let p ∈ AssRM and p = (0 :R m) = (0 :R c(m)), for some 0 6=m ∈ M. By [6, 1.2] c(m) is a nitely generated ideal of R; so suppose that c(m) = Rx1+· · ·+Rxt where xi ∈ R, for i = 1, . . . , t. Then it follows that p = (0 : Rx1, . . . , Rxt) =Tt
i=1(0 :Rxi)and sop= (0 :Rxj), for some1≤j≤t. Thus, p∈AssR.
To prove the last part, suppose that ρ : R → B is our structural ring homomorphism and p∈AssRR such thatp= (0 :R r), for somer ∈R. Then by [7, Lemma 3.1] (0 :R ρ(r)) = (0 :c(ρ(r))) which is equal to (0 :Rr) =p by the fact II just before Lemma 2.6. Since ρ(r)6= 0,p∈AssRB.
3. TRIVIAL RING EXTENSIONS BY CONTENT MODULES (ALGEBRAS)
Let R be a ring andM an R-module. The trivial ring extension ofR by M (also called the idealization of M over R) is the ring R0 := RαM whose underlying group is R×M, the set of pairs (r, m) with r ∈R,m ∈ M, with multiplication given by(r, m)(r0, m0) = (rr0, rm0+r0m). The idealization ofM over R is denoted byRoM.
In this section, we study the transfer of contentness of modules (algebras) to trivial ring extensions.
Lemma 3.1. Let M be anR-module and M0 :=RoM be the idealization of M over R and I an ideal of R. Then IM0 = (=ρ(I)M0) = IoIM where ρ:R→RoM is the natural mapping.
Proof. Let (r,Pk
i=1eimi)∈IoIM wherer∈I and ei∈I, mi ∈M, for i= 1, . . . , k. Then
(r,
k
X
i=1
eimi) = (r,0) + (0,
k
X
i=1
eimi) = (r,0)(1,0) +
k
X
i=1
(ei,0)(0, mi)∈IM0
and so IoIM ⊆IM0. The reverse inclusion is obvious.
Theorem 3.2. LetM be anR-module. ThenM is a contentR-module if and only if M0 :=RoM is a content R-module.
Proof. (→). Let M be an content R-module and (r, m) ∈ RoM. By denition and Lemma 3.1
cM0((r, m)) =∩{I :I is an ideal ofR and (r, m)∈IM0=IoIM}
=∩{I :I is an ideal of Randr ∈I, m∈IM} ⊇cM(m)∩Rr.
Since M is a content R-module, we have m∈cM(m)M ⊆cM0((r, m))M. Thus, (r, m)∈cM0((r, m))M0 =cM0((r, m))ocM0((r, m))M.
(←). Suppose that RoM := M0 is a content R-module and m ∈ M. Then (0, m) ∈ cM0((0, m))M0 = cM(m)M0 = cM(m)ocM(m)M, by Lemma 3.1. Hence, m∈cM(m)M.
Corollary 3.3. Let M be an R-module. Then M is a at content R- module if and only if RoM is a faithfully at content R-module.
Corollary 3.4. Suppose that B is anR-algebra such thatB is a at and content R-module. Then B0 :=RoB is a faithfully at R-algebra such that is a content R-module.
Acknowledgment. The author would like to thank Kharazmi University for the nancial support and the referee for his/her comments.
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Received 3 June 2011 Kharazmi University,
Faculty of Mathematical Sciences and Computer, 43 Mofateh Ave.,
Tehran, Iran taheri@khu.ac.ir
