The Uniqueness of a Certain Type of Subdirect Product

Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS.6(2)(2017), 62-76

Existence and Uniqueness of a Certain Type of Subdirect

Product

H. Khabazian, Department of Mathematical Science

Isfahan University of Technology, Isfahan, Iran1

Abstract. We introduce a ”l_F-type subdirect product” and show that every

ring is uniquely a lF-type subdirect product of a family of lC-simple rings.

Then,, we show some applications in maximal ring of quotients and Martindale’s

ring of quotients.

Keywords: lF-type subdirect product, ring of quotients, lC-ideal, lC-simple, r_AI-semiprime.

2010 Mathematics subject classification: 16S60, 16D25

Introduction

Any ring can be written as subdirect product of a family of rings in many ways.

For any setS of ideals of a ring R with zero intersection,R is isomorpic to a

subdirect product of the family{R/I|I ∈S}. But this representation is not

unique, even ifS is a set of certain ideals, unless certain types of subdirects is

considered.

1_{Corresponding author: [email protected]}

Received: 26 August 2016 Revised: 13 November 2016

In section 1 standard facts are collected and l_C-simple rings are introduced.

Then in section 2, lF-type subdirect product is introduced and we show that

every left faithful lAI-Noetherian ring is isomorphic to a lF-type subdirect

prod-uct of a finite family of l_C-simple rings and this representation is unique. In

section 3, applying maximal ring of quotients, we show some applications.

In this paper, for any setS of subgroups of an additive group we set Σ(S) =

X

I∈S

I andS is called independent ifX

I∈S

I is a direct sum.

For any class C of subgroups, a C-subgroup means a subgroup from the class

C, the class of minimalC-subgroups is shown byCmn _{and the class of maximal} C-subgroups is shown byCmx_.

For classesCandFof subgroups, the class of subgroups which are aC-subgroup

and aF-subgroup is shown byC ∩ F.

For an additive group M, the set of C-subgroups of M is shown by hC:Mi,

M is calledC-simpleif it has no properC-subgroup,M is called C-Artinian

ifM satisfies the descending chain condition onC-subgroups, in other words,

if whose C-subgroups satisfy DCC,M is called C-Noetherian if M satisfies

the ascending chain condition on C-subgroups, in other words, if whose C

-subgroups satisfy ACC, andM is called C-ind.finiteif every independent set

of C-subgroups is finite. For an additive group M and K ⊆ M, the set of

C-subgroups of M containing K is shown by hC ⊇Ki and we say that K is

completelyC-uniformifK6= 0 and for anyC-subgroupsI andL,I∩L= 0

implies K∩I = 0 or K∩L = 0. Finally, the class of completelyC-uniform

subgroups is shown byCcu_.

For any familyS of subsets of a set, we set Int(S) = \

I∈S

I and Un(S) = [

I∈S

I.

1

Preliminaries

Definition 1.1 LetR ba a ring andCbe a class of subgroups.

1. R is said to be C-semiprime if there exists no nonzero nilpotent C

-subgroup.

2. Ris said to beC-primeif for everyC-subgroupsIandJ,IJ = 0 implies

either I= 0 orJ = 0.

1. I⊆Ris said to be left inner faithfulifI∩annl(I) = 0.

2. I⊆Ris called aleft annihilatorif annl(annr(I)) =I.

3. An ideal which is a left annihilator is called aleft annihilator ideal.

4. For the subgroupsU andI, we set (U:I)l ={x∈R|xI ⊆U}.

In this paper, in the category of rings, the class of left inner faithful subgroups is

shown by lIF, the class of left inner faithful idealsIfor which annl(annl(I)) =I

is shown by lCand the class of left inner faithful subgroupsIfor which annl(I)

is also left inner faithful is shown by dlI_F. Also the class of ideals is shown by

I, the class of left ideals is shown by lI, the class of right ideals is shown by rI,

the class of left annihilator ideals is shown by lAI, the class of right annihilator

ideals is shown by r_AIand the class of left faithful subgroups is shown by l_F.

Thus, dlIF-ideals means a left inner faithful ideal I for which annl(I) is also

left inner faithful.

Recall that according to the terminologies used in this paper,_I∩lI_Findicates

the class of left inner faithful ideals, lCmn is the class of minimal lC-ideals, and

ΣhlCmn:Riis the sum of the minimal lC-ideals ofR.

It is good to know that since every l_C-ideal is a left annihilator ideal, we

have

I−Noetherian⇒rAI−Noetherian⇒lAI−Artinian⇒lC−Artinian

I−Noetherian⇒lAI−Noetherian⇒lC−Noetherian

Lemma 1.3 LetR be a ring. IfS is an independent set of right ideals andI

is a left annihilator, thenI∩Σ(S) = Σ{I∩J |J∈S}.

proof. We may assume that S has only two elements A and B. Suppose

v ∈ I∩(A⊕B). There exist a ∈ A and b ∈ B with v = a+b. For every

x∈annr(I) we haveax+bx=vx= 0, implyingax= 0. Thus,aannr(I) = 0,

implyinga∈annl(annr(I)) =I. Consequentlya∈I∩A. Similarly,b∈I∩B.

Lemma 1.4 LetRbe a ring. For any independent setS of left inner faithful

proof. For everyJ ∈Swe have annl(Σ(S))⊆annl(J), implying annl(Σ(S))∩

J = 0. Thus by Lemma 1.3,

annl(Σ(S))∩Σ(S) = Σ{annl(Σ(S))∩J|J ∈S}= 0

Lemma 1.5 LetRbe a ring. Any setS of left annihilator ideals in which for

every distinct elementsI, J ∈S we haveI∩J= 0, is independent.

proof. LetI∈S. SetT =S−{I}. We haveI∩Σ(T) = Σ({I∩J |J∈T}) = 0

by Lemma 1.3.

Lemma 1.6 LetRbe a ring.

1. An ideal K is a dlIF-ideal iff annl(K) is a left inner faithful ideal and

K⊆annl(annl(K)).

2. IfI is a dlIF-ideal, then annl(I) is a lC-ideal.

3. If I and J are left inner faithful ideals, then so are IJ and I∩J. Also

we have annl(IJ) = annl(I∩J).

4. IfI andJ are dlI_F-ideals, then so isI+J.

5. IfI andJ are lC-ideals, then so is I∩J.

proof. (1 and 2) Straightforward.

(3) It is easy to see that annl(IJ)∩J ⊆annl(I). Thus annl(IJ)∩(I∩J) = 0,

implying annl(I∩J)∩(I∩J) = 0, annl(IJ)∩(IJ) = 0 and annl(IJ) =

annl(I∩J).

(4) Since annl(I+J) = annl(I)∩annl(J), annl(I+J) is a lIF-ideal by (3).

On the other handI ⊆annl(annl(I))⊆annl(annl(I+J)) and similarly, J ⊆

annl(annl(I+J)), implyingI+J⊆annl(annl(I+J)). Applying (1) completes

the proof.

(5) annl(I)+annl(J) is a dlIF-ideal by (2) and (4). Thus annl(annl(I)+annl(J))

is a lC-ideal by (2). On the other hand,

annl(annl(I) + annl(J)) = annl(annl(I))∩annl(annl(J)) =I∩J

Lemma 1.7 LetRbe a ring andU be a l_C-ideal.

1. IfI is a left inner faithful ideal, then so isI/U, also we have

2. IfI is a dlI_F-ideal, then so is I/U, also we have

annl(annl(I/U)) = annl(annl(I∩annl(U))∩annl(U))/U.

Proof. (1) SetM = annl(U). M is a left inner faithful ideal and annl(M) =U.

So,IM is a left inner faithful ideal and

annl(I/U) = annl(I/annl(M)) = annl(IM)/annl(M) =

annl(IM)/U = annl(I∩M)/U

by Lemma 1.6. Thus, (I ∩annl(IM))M ⊆ IM ∩annl(IM) = 0, so (I ∩

annl(IM))⊆annl(M), implying (I/U)∩annl(I/U) = 0.

(2) I/U is a left inner faithful ideal and annl(I/U) = annl(I∩annl(U))/U

by (1). On the other hand I ∩annl(U) is a dlIF-ideal by Lemma 1.6, so

annl(I∩annl(U)) is a left inner faithful ideal, thus annl(I∩annl(U))/U is a

left inner faithful ideal. The rest is clear by (1).

Lemma 1.8 LetRbe a ring,U be a lC-ideal andI be an ideal containingU.

1. Iis a left inner faithful ideal iffI/U is so. In this case annl(I∩annl(U) +

U) = annl(I).

2. Iis a dlIF-ideal iffI/Uis so. In this case, annl(annl(I/U)) = annl(annl(I))/U.

3. I is a lC-ideal iffI/U is so.

Proof. (1⇒) Follows from Lemma 1.7.

(1⇐) We have (U:I)l∩I⊆U, thus annl(I)∩I⊆annl(U)∩U = 0.

SetJ = annl(I∩annl(U) +U). I∩annl(U) +U is a left inner faithful ideal

by Lemma 1.6 and Lemma 1.5, so (I∩annl(U) +U)∩J = 0, implyingI∩

annl(U)∩J = 0 and U ∩J = 0. Thus J ⊆annl(U), so I∩J = 0, implying

J ⊆annl(I). On the other hand, it is clear tat annl(I)⊆J.

(2⇒) Follows from Lemma 1.7.

(2⇐) I is a left inner faithful ideal, so annl(I/U) = annl(I∩annl(U))/U

by Lemma 1.7, thus annl(I∩annl(U)) is a left inner faithful ideal by (1).

Consequently,I∩annl(U) is a dlIF-ideal, implying thatA=U +I∩annl(U)

is a dlIF-ideal by Lemma 1.6. On the other hand annl(A) = annl(I) by (1), so

annl(I) is a left inner faithful ideal. ThereforeI is a dlIF-ideal. The rest can

be proven easily by applying (1) and Lemma 1.7.

Corollary 1.9 LetRbe a ring andU be a l_C-ideal. U is a maximal l_C-ideal

iffR/U is a lC−simple ring.

Lemma 1.10 LetRbe a ring.

1. IfS is a set of maximal lC-ideals with Int(S) = 0, then S=hlCmx:Ri.

2. hl_Cmn_{:Riis an independent set.}

Proof. (1) LetU be a maximal lC-ideal. There existsV ∈S with annl(U)∩

annl(V)6= 0, then annl(U) = annl(V), implyingU =V. ThusU ∈S.

(2) Follows from Lemma 1.5.

Lemma 1.11 LetRbe a ring with Int(hlCmx:Ri) = 0.

1. Every nonzero l_C-ideal contains a minimal l_C-ideal.

2. Every proper l_C-ideal is contained in a maximal l_C-ideal.

3. For every maximal lC-idealK, annl(K) = Int(hlCmx:Ri − {K}}).

Proof. (1) Let K be a l_C-ideal containing no minimal l_C-ideal. For every

maximal lC-idealJ we have K∩annl(J) = 0 by Lemma 1.6, implying K ⊆

annl(annl(J)) =J. ThusK⊆Int(hlCmx:Ri) = 0.

(2) Let J be a proper l_C-ideal. annl(J) is a nonzero lC-ideal, so contains a

minimal lC-ideal I by (1). Then J = annl(annl(J))⊆ annl(I), on the other

hand annl(I) is a maximal lC-ideal.

(3) annl(K) is a minimal lC-ideal, so for everyK6=L∈ hlCmx:Ri, annl(K)∩

L6= 0, implying annl(K)⊆L. So, annl(K)⊆Int(hlCmx:Ri − {K}}). On the

other handK∩Int(hlCmx:Ri − {K}}) = 0, implying Int(hlCmx:Ri − {K}})⊆

annl(K).

Proposition 1.12 LetR be a left faithful ring. The following are equivalent.

1. R is lC−Artinian.

2. R is lC−Noetherian.

3. R is lC−ind.finite.

4. hlCmn:Riis finite and every nonzero lC-ideal contains a minimal lC-ideal.

6. hl_Cmn_{:Riis finite and Σhl}

Cmn:Riis left faithful.

7. hlCmx:Riis finite and InthlCmx:Ri= 0.

In this case for any lC-idealK we haveK= InthlCmx⊇Ki.

Proof. (1⇔2) Follows from Lemma 1.6.

(2⇒3) Temporarily suppose that there exists an infinite independent set of

l_C-ideals. Then there exists an infinite independent set {Ji | i ≥ 1} of lC

-ideals. SetNn = annl(annl(P n

i=1Ji)). P n

i=1Ji is a dlIF-ideal by Lemma 1.6,

so Nn is a lC-ideal by Lemma 1.6. Also, Pni=1Ji ⊆ Nn by Lemma 1.6. On

the other hand we have Jn+1∩P n

i=1Ji = 0, then Jn+1 ⊆annl(P n

i=1Ji), so

Nn ⊆annl(Jn+1), implying Nn∩Jn+1 = 0. Thus Nn ⊂ Nn+1 for alln ≥ 1

which is a contradiction.

(3⇒2) Temporarily suppose that there exists a set {Nn | n≥1} of lC-ideals

withNn⊂Nn+1 for alln≥1. SetJn=Nn+1∩annl(Nn). Jn is a lC-ideal by

Lemma 1.6. On the other handJi ⊆Ni+1⊆Nn+1 for all 1≤i≤n, implying Pn

i=1Ji ⊆Nn+1. FurthermoreJn+1⊆annl(Nn+1), implyingJn+1∩ Pn

i=1Ji=

0. Thus {Jn | n ≥ 1} is an infinite independent set of lC-ideals which is a

contradiction.

(1 and 3⇒4) Follows from Lemma 1.11.

(4⇒5 and 6) We show that for any l_C-idealK we have K = Inthl_Cmx_⊇Ki.

Set J = InthlCmx⊇Ki. J is a lC-ideal by Lemma 1.6, so it is enough to

show that J ∩annl(K) = 0. Temporarily suppose that J ∩annl(K) 6= 0.

J∩annl(K) contains a minimal lC-idealIby Lemma 1.11, thenK⊆annl(I),

so annl(I)∈ hlCmx⊇Ki, thusJ ⊆annl(I), implying I ⊆annl(J) which is a

contradiction.

(5⇒1 and 2) It is obvious.

(6⇒4) Straightforward.

(6⇔7) Follows from Lemma 1.6.

Lemma 1.13 LetRba a ring,U be a left annihilator ideal andI⊆R.

1. (U:I)l= annl(annl(Iannr(U))).

2. IfU ⊆I andI/U is a left annihilator, then so isI.

Proof. (1) We have (U :I)lI ⊆ U, so (U :I)lIannr(U) = 0, implying (U :

annl(Iannr(U))I⊆annl(annr(U)) =U, implying annl(Iannr(U))⊆(U:I)l.

(2) SettingN = (U:I)rwe have, annr(I/U) =N/U so,

I/U = annl(annr(I/U)) = annl(N/U) = (U:N)l/U

implyingI= (U:N)l= annl(Nannr(U)) by (1).

Proposition 1.14 LetRbe a lAI−semiprime ring. For every lC-ideal U,R/U

is also a l_AI−semiprime ring.

Proof. LetI be an ideal containing U such that I/U is a nilpotent left

an-nihilator ideal. There exists n≥ 1 with (I/U)n _{= 0, implying I}n _{⊆U. I is}

a left annihilator ideal by Proposition 1.12, so is I∩annl(U). On the other

hand, (I∩annl(U))n ⊆In ⊆U and (I∩annl(U))n ⊆ annl(U)n ⊆annl(U),

so (I ∩ annl(U))n ⊆ U ∩annl(U) = 0, thus I ∩ annl(U) = 0, implying

I⊆annl(annl(U)) =U. ThusI/U = 0.

Proposition 1.15 LetR be a semiprime ring. For every lC-idealU, R/U is

also a semiprime ring.

Proof. LetIbe an ideal containingU such thatI/U is nilpotent. There exists

n≥1 with (I/U)n_{= 0, implying I}n_{⊆U. Thus (I∩ann}

l(U))n⊆In⊆U and

(I∩annl(U))n ⊆annl(U)n ⊆annl(U), so (I∩annl(U))n ⊆U ∩annl(U) = 0,

consequentlyI∩annl(U) = 0, implyingI⊆annl(annl(U)) =U. ThusI/U = 0.

Lemma 1.16 Any left faithful l_C-simple ring is indecomposable (as a ring).

Proof. LetRbe l_C-simple ring andAandB be ideals with R=A⊕B. We

have (B∩annl(B))R = (B∩annl(B))(A⊕B) = 0, soB∩annl(B) = 0. On

the othe hand, A⊆annl(B), implying annl(B) =A. Similarly, annl(A) =B.

Therefore,A= 0 orB= 0.

Lemma 1.17 LetR be a left faithful lC-simple ring. . For every lC-idealU,

U+ annl(U) is a left faithful ideal and contains ΣhrI∩lAIcu:Ri.

Proof. Let J be a completely lAI-uniform right ideal. Then, J ∩U = 0 or

J ∩annl(U) = 0, so J U = 0 or Jannl(U) = 0, implying J ⊆ annl(U) or

J ⊆annl(annl(U)) =U. Thus,J⊆U+ annl(U).

It worth to mention that every minimal right ideal, minimal ideal and

2

l

-type subdirect product

Definition 2.1 We use the notation S ⊆sd Y i∈I

Ri to indicate that the ringS

is a subdirect product of the family{Ri|i∈I} of rings. In this case:

1. For everyJ ⊆I we setSJ ={u∈S| ∀i6∈J, πi(u) = 0}

2. For everyj ∈Iwe set S∗j =πj(Sj)

3. If for every j ∈ I, S∗j is a C-subgroup of Rj, then we say that S is a

C-typesubdirect product of the family{Ri |i∈I}of rings and we write

S⊆C sd

Y

i∈I

Ri.

It is clear thatιj(S∗j) =Sj for every j∈I and X

i∈I

Si= M

i∈I

S∗i.

Lemma 2.2 Let{Ri|i∈I} be a family of rings andS⊆lsdF Y

i∈I

Ri.

1. For everyJ ⊆I, annl(SJ) =SI−J andSJ is a lC-ideal.

2. If A is a left faithful left ideal ofS, thenπj(A) is left faithful for every

j∈I.

Proof. (1) Straightforward.

(2) SetI = annl(πj(A)). We have IS∗j ⊆S∗j, soιj(IS∗j)⊆Sj ⊆S. On the

other hand,πj(ιj(IS∗j)A) =IS∗jπj(A)⊆Iπj(A) = 0, also for everyj6=i∈I,

πi(ιj(IS∗j)A) =πi(ιj(IS∗j))πj(A) = 0. So,ιj(IS∗j)A= 0, thusιj(IS∗j) = 0,

henceIS∗j = 0, implyingI= 0.

Lemma 2.3 Let{Ri|i∈I}be a family of lC−simple rings andS⊆lsdF Y

i∈I

Ri.

hl_Cmx_:Si={S

I−{j}|j∈I} andhlCmn:Si={Sj|j∈I}.

Proof. Let j ∈ I. SI−{j} is a lC-ideal by Lemma 2.2. On the other hand

S/SI−{j} ∼=Rj, thus S/SI−{j} is a lC−simple ring, consequently SI−{j} is a

maximal l_C-ideal by Corollary 1.9. Furthermore, Int({SI−{j} | j ∈ I}) = 0.

ThereforehlCmx:Si={SI−{j}|j∈I}by Lemma 1.10.

Proposition 2.4 Let{Ri|i∈I} be a family of rings andS⊆lsdF Y

i∈I

Ri.

2. S is semiprime iff for eachj∈I,Rj is semiprime.

Proof. (1)SI−{j}is a lC-ideal by Lemma 2.2, soS/SI−{j}is a lAI−semiprime

ring by Proposition 1.14. On the other hand we haveRj∼=S/SI−j.

(2⇒)SI−{j} is a lC-ideal by (2-2), soS/SI−{j} is a semiprime ring by

Propo-sition 1.15. On the other hand we haveRj∼=S/SI−{j}.

(2⇐). Let K ⊆ S be a an ideal with zero square. Then for every j ∈ I,

πj(K)2 = πj(K2) = 0 and πj(K) is an ideal, implying πj(K) = 0. Thus

K= 0.

Theorem 2.5 Let{Ri |i∈I} and{Qu|u∈A} be a families of lC−simple

left faithful rings, S ⊆lF

sd Y

i∈I

Ri and T ⊆lsdF Y

u∈A

Qu. If φ : S −→ T is an

isomorphism, then there exists a bijectionf :I −→A and for eachi ∈I, an

isomorphismφi:Ri−→Qf i) such that onS we haveφ=Qi∈Iφi.

Proof. Follows from Lemma 2.3 we have

hl_Cmx_:Si={S

I−{j}|j∈I}andhlCmx:Ti={TA−{u}|u∈A}

So there is a bijection f : I −→ A such that for each i ∈ I, φ(SI−{i}) =

TA−{f(i)}. The mapφi:Ri−→Qf(i)defined byφi(πi(x)) =πf(i)(φ(x)) is well

defined and an isomorphism.

Theorem 2.6 Let R be a nonzero ring withhl_Cmx_{:Ri= 0. R is a l}

F−type

subdirect product of a family of lC−simple rings and this representation is

unique.

Proof. We may havehl_Cmx_:Ri={U

i |i∈I} for some set I. Set Ri =R/Ui

and consider the homomorphism Φ :R−→Q

i∈IRi given by Φ(r) ={r+Ui|

i∈I}. Now set S= Φ(R). Clearly Φ is a monomorphism. For each j ∈I we

setVj=∩j6=i∈IUi. We have

Sj={ιj(r+Uj)|r∈Vj}

On the other hand we have Vj = annl(Uj) by Lemma 1.11. Thus S∗j =

annl(Uj)/Uj, implying annl(S∗j) = annl(annl(Uj))/Uj = 0. Finally Rj is a

lC−simple ring by Corollary 1.9.

Theorem 2.7 Any left faithful lC−Noetherian ring is isomorphic to a lF−type

subdirect product of a finite family of l_C−simple rings and this representation

Proof. Follows from Theorem 2.6 and Proposition 1.12.

Theorem 2.8 Any l_AI−semiprime l_C-Noetherian ring is isomorphic to a

lF−type subdirect product of a finite family of lAI−prime rings and this

representation is unique.

Proof. It is easy to see that a ring is lAI−prime iff it is lAI−semiprime and

lC-simple. Applying Theorem 2.7 and Lemma 2.4 completes the proof.

Theorem 2.9 Any semiprime lC-Noetherian ring is isomorphic to a lF−type

subdirect product of a finite family of prime rings and this representation is

unique.

Proof. It is easy to see that a ring is prime iff it is semiprime and l_C-simple.

Applying Theorem 2.7 and Lemma 2.4 completes the proof.

3

Applications

In [3, (13.21)], it is shown that the maximal right ring of quotients of a left

faithful ringRexists and is the unique ring extensionQofRsuch that

1. For every dense right idealLofRand anyR-homomorphismf :L−→R

there existsq∈Qsuch thatf(x) =qxfor allx∈L.

2. For everyq∈Qthere exists dense right idealLofRwithqL⊆R.

3. Every dense right ideal I ofR is left Q-faithful (q ∈Q, qI = 0 implies

q= 0).

Also, this extension is shown by Qr

max(R). Therefore, by this description, we

have Proposition 3.5. But before that we need some lemmas.

For a ring R, I ⊆ R and l ∈ R we set (I :l)r = {r ∈ R | lr ∈ I}. Pay

attention that for any I, J ⊆ R, r(J :r)r ⊆ J, so r(J :r)rI ⊆ J I, implying

(J:r)rI⊆(J I:r)r. Thus, for any dense right idealJ and any left faithful right

idealI,J I is a dense right ideal.

Lemma 3.1 LetQ be a ring andR be a subring ofQsuch that every dense

right ideal of R is left Q-faithful. For any right ideal L ⊆ R and any

R-homomorphismf :L−→Q, iff(I∩L) = 0 for a dense right idealIofR, then

Proof. For everyl∈L,f(l)(I:l)r=f(l(I:l)r)⊆f(I∩L) = 0, sof(l) = 0.

Lemma 3.2 Let R be a ring and S be a subring of R. If N ⊆ S is a left

R-faithful ideal ofR, then for any dense right idealJofS,J N is a dense right

ideal ofR.

Proof. Let r ∈ R. First we show that (J :r)r is left R-faithful. Suppose

t(J:r)r= 0 for a t∈R. Then, for everyu∈N, tu(J:ru)r= 0, so tu(S∩(J:

ru)r) = 0, implying tu = 0, Thus, tN = 0, so t = 0. On the other hand

(J:r)rN ⊆(J N:r)r, so (J N:r)ris left faithful.

Lemma 3.3 Let Q be a ring and R be a subring of Q such that for every

q ∈ Q, there exists a left Q-faithful L ⊆ R with qL ⊆ R. If R is l_C-simple,

then so isQ.

Proof. Firs we show that for every right idealAofQ, annl(R∩A) = annl(A)

in Q. Let q ∈ A. There exists a left Q-faithful L ⊆ R such that qL ⊆ R.

Then,qL⊆R∩A, so annl(R∩A)qL= 0, implying annl(R∩A)q= 0. Thus,

annl(R∩A)A= 0. It is easy to see that for every right idealAofQ,R∩A= 0

impliesA= 0. Now let AandB be disjoit ideals ofQwith annl(A) =B and

annl(B) =A. SetI =R∩A andJ =R∩B. I andJ are disjoit ideals ofR.

Also in R, annl(I) =R∩B =J and annl(J) = R∩A =I. Thus, I = 0 or

J = 0, implyingA= 0 orB= 0.

Lemma 3.4 Let{Ri|i∈I} be a family of left faithful rings. SetS= Y

i∈I

Ri.

1. If Kj is a dense right ideal of Rj for eachj ∈I, then M

i∈I

Ki is a dense

right ideal ofS.

2. If L is a dense right ideal of S, then L∗j and so πj(L) is a dense right

ideal ofRj for eachj∈I.

Proof. (1) Let x, y ∈ S and y(M

i∈I

Ki:x)r = 0. For each j ∈ I we have,

ιj((Kj:πj(x))r)⊆( M

i∈I

Ki:x)r, soπj(y)(Kj:πj(x))r= 0, implying πj(y) = 0.

Thus,y= 0.

(2) Let x, y ∈ Rj and y(L∗j:x)r = 0. We have (L:ιj(x))r = Y

i∈I

Ni, where

Nj = (L∗j :x)r and Ni = Ri for j 6= i ∈ I. Thus, ιj(y)(L:ιj(x))r = 0, so

Proposition 3.5 Let R be a left faithful ring and S be a subring of R, If S

contains a leftR-faithful ideal ofR, then Qrmax(S) = Qrmax(R).

Proof. S contains a leftR-faithful idealN ofR. SetQ= Qr

max(R). First let

J be a dense right ideal ofS andqJ = 0 for aq∈Q. J N is a dense right ideal

of S by Lemma 3.2 and qJ N = 0, implying q = 0. Thus, every dense right

ideal ofS is leftQ-faithful.

Now letJ be a dense right ideal of S andf :J −→S be aS-homomorphism.

SetL=J N,L is a dense right ideal ofR contained inJ by Lemma 3.2. Let

x ∈ L and r ∈ R. For every u ∈ N, f(x)ru = f(xru) = f(xr)u, Hence,

(f(x)r−f(xr))N = 0, implying f(x)r = f(xr). Thus, f : L −→ S is an

R-homomorphism, so there exists q ∈ Q such that f(x) =qx for all x ∈ L.

The map g : J −→ Q given by g(x) = f(x)−qx is a S-homomorphism and

g(L) = 0. On the other hand, Lis dense right ideal of S, sog= 0 by Lemma

3.1. Thus,f(x) =qxfor allx∈J.

Finally, letq∈Q. There exists dense right idealL ofR with qL⊆R. Thus,

qLN ⊆RN ⊆S. On the other hand,LN is a dense right ideal of R, so is a

dense right ideal ofS.

Lemma 3.6 Let{Ri|i∈I}be a family of left faithful rings. Then, Qrmax( Y

i∈I

Ri) =

Y

i∈I

Qrmax(Rj).

Proof. SetS=Y

i∈I

RiandQ= Y

i∈I

Qr

max(Rj). First, letLbe a dense right ideal

ofSandf :L−→Sbe aS-homomorphism. SetN =L(P

i∈ISi). Nis a dense

right ideal ofS andN =P

i∈ILSi. Also, for every j∈I,f(LSj) =f(L)Sj⊆

Sj andπj(LSj) is a dense right ideal of Rj by Lemma 3.4. Consider the well

definedRj-homomorphismfj:πj(LSj)−→Rj given byfj(πj(x)) =πj(f(x)).

There existsqj∈Qrmax(Rj) such thatfj(πj(x)) =qjπj(x) for allx∈LSj. Set

q={qi|i∈I}. Then,f(x) =qxfor allx∈N. Thus,f(x) =qxfor allx∈L

by Lemma 3.1.

Now letq∈Q. For eachj∈I, there exists a dense right idealKj ⊆Rj such

thatπj(q)Kj⊆Rj. Thus,q M

i∈I

Ki⊆S. On the other hand, M

i∈I

Ki is a dense

right ideal ofS by Lemma 3.4.

Finally, let Lbe a dense right ideal of S, q ∈Qand qL = 0. Then, for each

j∈I,πj(q)πj(L) = 0, implyingπj(q) = 0 becauseπj(L) is a dense right ideal

Proposition 3.7 Let{Ri|i∈I}be a family of rings andS ⊆lsdF Y

i∈I

Ri. Then,

Qr

max(S) = Y

i∈I

Qr

max(Rj).

Proof. We haveM

i∈I

S∗j= X

j∈I

Sj, so M

i∈I

S∗j is a left faithful ideal ofS and a

left faithful ideal ofY

i∈I

S∗j. Thus,

Qr_max(S) = Qr_max(M

i∈I

S∗j) = Qrmax( Y

i∈I

S∗j) = Y

i∈I

Qr_max(S∗j) = Y

i∈I

Qr_max(Rj)

by Proposition 3.5 and Lemma 3.6.

In [3, (14.7)], Martindale’s right rings of quotients is introduced for semiprime

rings. We can extend this definition to any left faithful ringR as follow.

Qr(R) ={q∈Q_maxr (R)|qA⊆R for some left faithful idealA}

Proposition 3.8 Let{Ri|i∈I}be a family of rings andS ⊆lsdF Y

i∈I

Ri. Then,

Qr_{(S) =}Y i∈I

Qr_(R i).

Proof. Letq∈Qr_{(S). Then q∈Q}r

max(S) and qA⊆S for some left faithful

idealAof S. Letj∈I. πj(A) is a left faithful ideal of Rj by Lemma 2.2, on

the other hand, πj(q)πj(A) ⊆ Rj, so πj(q)∈ Qr(Rj). Thus, q ∈ Y

i∈I

Qr(Ri).

Now let q ∈Y i∈I

Qr_(R

i). Letj ∈I and set qj =πj(q). Then, qj ∈Qrmax(Rj)

andqjKj ⊆Rj for a left faithful idealKj ofRj. KiS∗j is also a left faithful

ideal of Rj. and qjKiS∗j ⊆S∗j. Thus, A=Pj∈Iιj(KjS∗j) is a left faithful

ideal ofS andqA⊆S. Therefore,q∈Qr_(S).

Corollary 3.9 For any l_C-simple ring R, Qr

max(R) and Qr(R) are also lC

-simple.

Proof. Follows from Lemma 3.3.

In this order, applying Theorem 2.7, Proposition 3.7, Proposition 3.8,

Corol-lary 3.9 and Lemma 1.16, provide us with the decomposition of the maximal

right ring of quotients and Martindale’s right ring of quotients of any left left

Theorem 3.10 Any left faithful l_C-Noetherian ring R having no proper left

faithful ideal which contains ΣhrI∩lAIcu:Riis isomorphic to a direct product

of a finite family of lC-simple rings Ri with no proper left faithful ideal which

contains Σhr_I∩l_AIcu_:R

iiand this representation is unique.

Proof. LetRbe a left faithful lC-Noetherian ring with no proper left faithful

ideal which contains ΣhrI∩lAIcu:Ri . We have R ∼= S ⊆lF

sd Qn

i=1Ri, where

each Ri is a lC-simple ring by Theorem 2.7. On the other hand, for each

1≤j ≤n, SI−{j}+Sj is a left faithful ideal and contains ΣhrI∩lAIcu:Siby

Lemma 1.17, soS=SI−{j}+Sj, implyingπj(Sj) =Rj. Thus,S=Q n

i=1Ri.

Theorem 2.9 is obtained in [1, Theorem 17 and Theorem 19] in a long process.

rAI−semiprime property has been taken under consideration in [2, Proposition

2.7 and Theorem 2.8] and can be characterized as follow:

Proposition 3.11 LetRbe a l_I−Artinian ring.

1. Z(RR) is the nilpotent right annihilator ideal containing all nilpotent

right annihilator ideals.

2. R is left nonsingular iffR is r_AI−semiprime.

3. R/Z(RR) is left nonsingular and rAI−semiprime.

References

[1] H. Khabazian, THE STRUCTURE OF RINGS IN WHICH THE

INTER-SECTION OF INESSENTIAL PRIME IDEALS IS ZERO, Bulletin of the

Iranian Mathematical Society, Vol. 25, No. 1 (1999), 1-23.

[2] H. Khabazian, SOME CHARACTERIZATIONS OF ARTINIAN RINGS,

International Electronic Journal of Algebra, Vol. 9 (2011), 1-9.

[3] T. Y. Lam, Lectures on Modules and Rings, Spring-Verlag, New York,