REND.SEM.MAT.UNIV.PADOVA, Vol. 120 2008)
Finitely Presented Modules over Right Non-Singular Rings.
ULRICH ALBRECHT *)
ABSTRACT - Thispaper characterizesthe right non-singularrings R for which M=ZM) isprojective whenever M isa cyclically finitely) presentedmodule. Several related results investigate right semi-hereditary rings.
1. Introduction.
The straightforward attempt to extend the notion of torsion-freeness from integral domainsto non-commutative ringsencountersimmediate difficulties. To overcome these, one can concentrate on either the compu- tational or the homological propertiesof torsion-free modules.Goodearl and otherstook the firstapproach when they introduced the notion of a non- singular module [8]. A right R-module M is non-singular if ZM) 0 where ZM) fx 2 M j xI 0 for some essential right ideal I of R} denotesthe singular submodule of M. On the other hand, M is singular if ZM) M. Moreover, a submodule U of an R-module M is S-closed if M=U isnon- singular. Finally, R isa right non-singular ring if RR isnon-singular. The right non-singular rings are precisely the rings which have a regular, right self-injective maximal right ring of quotients, which will be denoted by Qr see [8] and [11] for details). Following [11, Chapter XI], Qr is a perfect left localization of R if Qr isflat asa right R-module and the r r r r multiplication map Q R Q ! Q is an isomorphism. In particular, Q isa perfect left localization of R if and only if every finitely generated non- singular right R-module can be embedded into a projective module [8] and [11]). We call a right non-singular ring with this property right strongly non-singular.
*) Indirizzo dell'A.: Department of Mathematics, Auburn University, Auburn, AL 36849, U.S.A. E-mail: [email protected] 46 Ulrich Albrecht
Hattori took the second approach by defining M to be torsion-free if R Tor1 M; R=Rr) 0 for all r 2 R [9]. The classes of torsion-free and non- singular right R-modulescoincide if and only if R isa right Utumi p.p.-ring without an infinite set of orthogonal idempotents [3, Theorem 3.7]. Here, R isa right p.p.-ring if all principal right idealsof R are projective. Moreover, a right non-singular ring R isa right Utumi-ring if every S-closed right ideal of R isa right annihilator. Closely related to the notion of torsion-freeness are those of purity and relative divisibility. A sequence of right R-modulesis pure-exact RD-exact) if every finitely presented cyclically presented) module is projective with respect to it. Investigating RD- and pure-projective modules leads to the investigation of the condition that M=ZM) isprojective. The dual question when ZM) isinjective hasbeen addressedin [8, Page 48, Example 24]. Section 2 discusses the question for which rings M=ZM) isprojective for all RD-projective modules M. Theorem 2.1 shows that, provided R hasno in- finite set of orthogonal idempotents, these are precisely the right Utumi p.p.- rings discussed in [3]. The structure of pure-projective right R-moduleswas described in [4] in case that R is a right strongly non-singular, right semi- hereditary ring R without an infinite set of orthogonal idempotents. Theo- rem 2.3 shows that these conditions on R are not only sufficient, but also necessary for the structure-theorem part b) of Theorem 2.3 to hold. PruÈfer domainscan be characterized asthe domainswith the property that, whenever a torsion-free module M containsa projective submodule U with M=U finitely generated, then M isprojective and M=U isfinitely presented [7, Chapter VI]. We show that the right non-singular rings having the corresponding property for non-singular modules are precisely the right strongly non-singular, right semi-hereditary rings of finite right Goldie dimension Theorem 3.1). Section 4 investigates pure-projective modules over right hereditary rings. As part of our discussion, we obtain a characterization of the right Noetherian right hereditary ringswith the restricted right minimum condition which are right strongly non-singular. The last results of this r paper demonstrate that right invertible submodules of QR which were introduced in [11, ChaptersII.4 and IX.5] may fail to sharemany of the important propertiesof invertible modulesover integral domains.For instance, the lattice of finitely generated right ideals over a PruÈfer domain isdistributive, i.e. I \ J K) I \ J) I \ K) for all finitely generated ideals I, J, and K of R [7, Theorem III.1.1]. Example 4.7 shows that there exists a right strongly non-singular, hereditary, right and left Noetherian ring whose finitely generated right ideals do not have this property. Finitely Presented Modules over Right Non-Singular Rings 47
2. RD-Projective Modules.
Let U be a submodule of a non-singular module M. The S-closure of U in M isthe submodule V of M which contains U such that V=U ZM=U).
THEOREM 2.1. The following are equivalent for a right non-singular ring R without an infinite set of orthogonal idempotents: a) R is a right Utumi p.p.-ring. b) M=ZM) is projective for every RD-projective module M.
PROOF. a) ) b): Since every RD-projective module isa direct sum- mand of a direct sum of cyclically presented modules, it suffices to verify b) in case that M R=aR for some a 2 R. Let J be the right ideal of R which contains aR such that J=aR ZR=aR). Then, R=J M=ZM) isa non- singular cyclic module which is projective by [3, Corollary 3.4]. b) ) a): Let I be the S-closure of rR for some r 2 R. Since R=rR isRD- projective, and R=I R=rR)=ZR=rR), we obtain that R=I isprojective. Thus, I isgenerated by an idempotent. By [3, Lemma 3.5], it suffices to show that every S-closed right ideal J of R isgenerated by an idempotent. For this,select 0 6 r0 2 J. Since J is S-closed in R,itcontainstheS-closure I0 of r0R.Bywhat hasbeen shown sofar, I0 e0R for some idempotent e0 of R.Hence, J e0R È [J \ 1 À e0)R]. If J \ 1 À e0)R 6 0, select a non-zero r1 1 À e0)r1 2 J; and observe that J \ 1 À e0)R is S-closed in R. Hence, it containsthe S-closure I1 of r1R in R.Bytheprevious paragraph, I1 fR for some idempotent f of R.Writef 1 À e0)s for some s 2 R,andsete1 f 1 À e0). Since e0 f 0, we have 2 2 2 2 e1e0 e0e1 0ande1 f À f e0 À fe0 f fe0) f À fe0 e1.Thus,e0 and e1 are non-zero orthogonal idempotentswith e1R fR.Onthe other hand, f f 1 À e0)s e1s yields fR e1R. Consequently, R e0R È e1R È [J \ 1 À e0 À e1)R]. Continuing inductively, we can construct non-zero orthogonal idempotents e0; ...; en1 2 J aslong as J \ 1 À e0 À ...À en)R 6 0. Since R doesnot contain an infinite family of orthogonal idempotent, thisprocesshasto stop, say
J \ 1 À e0 À ...À en)R 0. Then, e0 ... em isan idempotent with J e0 ... em)R. p
We now investigate which conditions R has to satisfy to ensure the validity of the structure theorem for pure-projectives in [4]. We want 48 Ulrich Albrecht to remind the reader that a right R-module is essentially finitely generated if contains an essential, finitely generated submodule.
LEMMA 2.2. The following are equivalent for a right non-singular ring R: a) R is right semi-hereditary and has finite right Goldie-di- mension. b) A finitely generated right R-module M is finitely presented if and only if p:d:M 1.
PROOF. a) ) b): Since R isright semi-hereditary, every finitely pre- sented module has projective dimension at most 1. Conversely, whenever M Rn=U for some projective module U, then U isessentially finitely generated since R has finite right Goldie dimension. By Sandomierski's Theorem [5, Proposition 8.24], essentially finitely generated projective modulesare finitely generated. b) ) a): Clearly, R hasto be right semi-hereditary. If R hasinfinite right Goldie-dimension, then it contains a family fIngn < v of non-zero, fi- nitely generated right ideals whose sum is direct. Since R isright semi- hereditary, each In isprojective, and the sameholdsfor Èn THEOREM 2.3. The following conditions are equivalent for a right non- singular ring R without an infinite set of orthogonal idempotents: a) R is a right Utumi), right semi-hereditary ring such that Qr is a perfect left localization of R. b) A right R-module M is pure-projective if and only if i) ZM) is a direct summand of a direct sum of finitely gen- erated modules of projective dimension at most 1. ii) M=ZM) is projective. PROOF. A right strongly non-sigular, right semi-hereditary ring without an infinite set of orthogonal idempotents is a right Utumi ring [3, Theorems3.7 and 4.2]. a) ) b): By [3], R has finite right Goldie-dimension. Because of Lemma 2.2, ZM) ispure-projective if and only if condition i) in b) holds.It remains to show that M=ZM) isprojective whenever M isfinitely presented. However, thisfollowsfrom [11] since M=ZM) isa finitely generated non- singular module, and Qr isa perfect left localization of R. Finitely Presented Modules over Right Non-Singular Rings 49 b) ) a): To see that R is right semi-hereditary, consider a finitely generated right ideal I if R. The S-closure J of I satisfies J=I ZR=I). Hence, R=J isprojective by b), and J=I has projective dimension at most 1. Since J isprojective, thisisonly possibleif I isprojective. To show that R hasfinite right Goldie-dimension, consider a right ideal of R of the form a0R È ...È anR È ...where each an 6 0. For m < v, let Im be the S-closure of a0R È ...amR. Since R=Im isprojective by b), Im isgen- erated by an idempotent em of R. Write Im1 Im È [Im1 \ 1 À em)R]. Observe that [Im1 \ 1 À em)R] isgenerated by an idempotent f of R asa direct summand of R. Setting dm f 1 À em) yieldsan idempotent dm of R such that emdm dmem 0, and Im1 Im È dmR asin the proof of The- orem 2.1. Inductively, one obtainsan infinite family of orthogonal idempo- tents fdmjm < vg of R, which is not possible. Thus, R hasfinite right Goldie- dimension; and every S-closed right ideal J of R isthe S-closure of a finitely generated right ideal. By b), R=J isprojective; and R isa right Utumi-ring since J eR for some idempotent e of R. To establish that Qr isa perfect left localization of R, it suffices to show that every finitely generated non-singular right R-module M isprojective. Write M Rn=U and observe that U is essentially finitely generated. Select a finitely generated essential submodule V of U. Then, U=V ZRn=U), and M isprojective by b). p In the following, the injective hull of a module M isdenoted by EM). COROLLARY 2.4. Let R be a right semi-hereditary ring of finite right Goldie-dimension such that Qr is a perfect left localization of R. A right R- module M is pure-projective if and only if M=ZM) is projective and ZM) is isomorphic to a direct summand of a direct sum of finitely generated submodules of Qr=R)n. PROOF. We first show that a finitely generated singular module M has projective dimension at most 1 if and only if it can be embedded into a finite direct sum of copies of Qr=R.Ifp:d:M 1, then there exist a finitely generated free module F and an essential projective submodule P of F with M F=P. Since R hasfinite right Goldie-dimension, P is essentially fi- nitely generated, and hence itself finitely generated by Sandomierski's Theorem [5, Proposition 8.24]. We can find a finitely generated projective module U such that P È U isa finitely generated free module, say P È U Rn. Since M issingular, P È U is an essential submodule of F È U. Therefore, F È U EP È U) Qr)n, and M Qr=R)n. 50 Ulrich Albrecht On the other hand, if M isa finitely generated submodule of Qr=R)n, then there isa finitely generated submodule U of Qr)n containing Rn such that M U=Rn. Since R is right semi-hereditary, and since Qr isa perfect left localization of R, U isprojective and p:d:M 1. The corollary now followsdirectly from Theorem 2.3. p 3. Essential Extensions of Projective Modules. In the commutative setting, PruÈfer domainsare characterized by con- ditionsb) and c.ii) [7]. THEOREM 3.1. The following are equivalent for a right non-singular ring R: a) R is a right semi-hereditary ring of finite right Goldie-dimen- sion for which Qr is a perfect left localization of R. b) Whenever a non-singular module M contains a projective submodule U such that M=U is finitely generated, then M is projective and M=U is finitely presented. c) i) R is a right p.p.-ring. ii) If a finitely generated non-singular right R-module M con- tains an essential projective submodule U, then M is projective, and M=U is finitely presented. PROOF. a) ) b): Let W be the S-closure of U in M. Since M=W isfi- nitely generated asan image of M=U and non-singular, it is projective by a). Hence, M W È P for some finitely generated projective module P.We may thusassumethat M=U issingular. Since R isright semi-hereditary, U ÈIUi where each Ui isfinitely generated [1]. Because M=U issingular and M isnon-singular, U is essential in M.Thus,EM) EU) ÈIEUi) in view of the fact that direct sums of non-singular injectives are injective if R hasfinite right Goldie-dimension [11, Proposition XIII.3.3]. Choose a finitely generated submodule V of M such that M U V.ThereisafinitesubsetJ of I such that V ÈJ EUi). Then, W1 V ÈJUi isa finitely generated submodule of ÈJEUi) such that V \ ÈInJ Ui) 0. Consequently, M W1 ÈÈInJUi.ButW1 isprojective by a) showing that M ispro- jective and that M=U W1=U isfinitely presented. b) ) c): Observe that every finitely generated non-singular module is projective by choosing U 0 in b). Finitely Presented Modules over Right Non-Singular Rings 51 c) ) a): Assume that R containsa right ideal U of the form U Èn A submodule U of a module M is tight if both U and M=U have pro- jective dimension at most 1. A module is coherent if all itsfinitely gener- ated submodules are finitely presented. COROLLARY 3.2. Let R be a right semi-hereditary ring of finite right Goldie-dimension such that Qr is a perfect localization of R. a) A right R-module M of projective dimension at most 1 is co- herent. Moreover, all its finitely generated submodules are tight. b) If M is singular and a direct sum of countably generated modules, then p:d:M 1 if and only if M Qr=R)I) for some index-set I. PROOF. a) Write M F=P where F and itssubmodule P are projec- tive. If U isa finitely generated submodule of M, then there isa submodule W of F containing P with W=P U. By Theorem 3.1, W isprojective and W=P isfinitely presented.Clearly, U and M=U have projective dimension at most 1. b) Without loss of generality, we may assume that M iscountably generated. If p:d:M 1, then M F=P where F isprojective and P RI) for some index-set I. Since P is essential in F, we have F EP) Qr)I) 52 Ulrich Albrecht by [11, Proposition XIII.3.3] because R hasfinite right Goldie-dimension. Hence, M Qr=R)I). Conversely, suppose that M Qr=R)v), and select a submodule U of r v) v) v) Q ) containing R such that M U=R . Choose fun jn < vgU v) v) ` such that U Sn 4. Hereditary Rings. The first result describes the right strongly non-singular, right Noe- therian, right hereditary rings. PROPOSITION 4.1. The following conditions are equivalent for a right non-singular ring R of finite right Goldie dimension: a) R is a right strongly non-singular, right hereditary ring with- out an infinite set of orthogonal idempotents. b) R is a right strongly non-singular, right Noetherian and right hereditary. c) i) R has finite right Goldie dimension. ii) M is pure projective if and only if M=ZM) is projective, and ZM) is a direct summand of a direct sum of finitely generated modules. PROOF. a) ) b): By [3, Theorems3.7 and 4.2], R hasfinite right Goldie dimension. However, essentially finitely generated projective modules are finitely generated [5, Proposition 8.24]. b) ) c) isobviousin view of Theo- rem 2.3. c) ) a): Let I be a right ideal of R, and J its S-closure in R. Since R has finite right Goldie dimension, I containsa finitely generated right ideal K as an essential submodule. Thus, J isthe S-closure of K, and J=K isthe singular submodule of the finitely presented module R=K. By c), R=J is projective, and R J È J1. Then, R=I J=I È J1. In particular, J=I isa finitely generated singular module, which is pure-projective by c). Hence, J=I is a direct summand of a direct sum of finitely presented modules. Clearly, thissumcan be chosento be finite. Therefore, J=I isfinitely presented, and I isfinitely generated since J isa direct summand of R. Once we have shown that every finitely generated non-singular right R- Finitely Presented Modules over Right Non-Singular Rings 53 module M isprojective, we will have establishedthat R isa right heredi- tary ring with the property that Qr isa perfect left localization of R. There exists a finitely generated free module F and a submodule U of F such that M F=U. Since R hasfinite right Goldie-dimension, U contains a finitely generated essential submodule V. Because, F=U isnon-singular, U=V is the singular submodule of the finitely presented module F=V.By c), F=U F=V)=U=V) isprojective. p COROLLARY 4.2. The following are equivalent for a right non-singular ring R with finite right Goldie-dimension: a) R is a right Noetherian, right hereditary ring which satisfies the restricted right minimum condition such that Qr is a perfect left lo- calization of R. b) M is pure projective if and only if M=ZM) is projective and ZM) is a direct summand of a direct sum of finitely generated Artinian modules. PROOF. a) ) b): Since R hasthe restrictedminimum condition, every finitely generated singular right module is Artinian. b) ) a): Let I be an essential right ideal of R.SinceR hasfinite right Goldie-dimension, I contains a finitely generated essential right ideal J. By c), the finitely presented module R=J isa direct summand of a finite) direct sum of finitely generated Artinian modules. But this is only pos- sible if R=J isArtinian. But then, R=I isArtinian too. Arguing asin the proof of Proposition 4.1, we obtain that R isa right stronglynon-singular, right hereditary. p By [8, Proposition 5.27], a right hereditary, right Noetherian ring which isthe product of prime ringsand ringsMorita equivalent to lower trian- gular matrix rings over a division algebra is right strongly non-singular and hasthe restrictedright minimum condition, i.e. R=I isArtinian for every essential right ideal I of R. r r Let U be a subset of Q , and set R : U)r fq 2 Q j Uq Rg and r R : U)` fq 2 Q j qU Rg. THEOREM 4.3. The following are equivalent for a ring R: a) R is a right Noetherian right hereditary ring satisfying the restricted right minimum condition such that Qr is a perfect left locali- zation of R. 54 Ulrich Albrecht b) R is a left Noetherian left hereditary ring satisfying the re- stricted left minimum condition such that Q` is a perfect right localization of R. c) R is a right and left Noetherian, hereditary, right and left Utumi-ring. d) R R1  ... Rn where each Ri is either a prime right and left Noetherian hereditary ring or Morita equivalent to a lower triangular matrix ring over a division algebra. PROOF. a) ) c): By [3, Theorem 4.2], Qr Q` is a semi-simple Artinian ring, and R isright and left Utumi. Furthermore, R isleft semi-hereditary by [3, Theorem 5.2]. It remainsto showthat R isleft Noetherian. Suppose that R containsa left ideal I which isnot finitely gen- erated. Without loss of generality, we may assume that I is essential in R.SinceQr Q` is semi-simple Artinian, R hasfinite left Goldie- dimension [11, Theorem XII.2.5], and I containsa finitely generated essential left ideal J0.SinceI isnot finitely generated, we can find an ascending chain J0 ... Jn ... of finitely generated essential left idealsinside I with Jn 6 Jn1 for all n. Since Qr isan injective left R-module being the maximal left ring of r quotientsof R,everymapf : Ji ! Q isright multiplication by some r q 2 Q , which isuniquely determined by f since Ji isessential. à Therefore, we can identify HomRJi; R)andJi R : Ji)r . Moreover, à Ji isa finitely generated projective right R-module because Ji is Ãà à projective since R isleft semi-hereditary.Furthermore, Ji R : Ji )` Ãà Ãà satisfies Ji Ji. To see this, observe that Ji Ji by definition. Conversely, Ji hasa finite projective basiss ince it isfinitely generated à and projective. There are a1; ...; ak 2 Ji and q1; ...; qk 2 Ji such that r y yq1a1 ... yqkak for all y 2 Ji.SinceQ also is the maximal left ring of quotientsof R, it isnon-singular asa left R-module. Because Ji Ãà is an essential left ideal, 1 q1a1 ... qkak.Ifz 2 Ji ,thenzqi 2 R, and z zq1)a1 ...zqk)ak 2 Ji. à à à We obtain a descending chain J0 ... Jn ... R R of finitely r r à generated submodules of QR. Since Q =R issingular, J0=R isa finitely generated singular right R-module. By the restricted right minimum à condition for R, J0=R isArtinian. Consequently, there is m with à à Jm Jm1, from which one obtains Jm Jm1. c) ) d): By [3, Theorem 5.2], the classes of torsion-free, non-singular and flat right R-modulescoincide. Becauseof [3, Theorem 5.5], R hasthe desired form. Finitely Presented Modules over Right Non-Singular Rings 55 d) ) a): By [3, Theorem 5.5], R isa right and left Noetherian heredi- tary ring for which the classes of torsion-free, flat and non-singular mod- ulescoincide. Thus, Qr isa perfect left localization of R by [3, Theorem 5.2]. Finally, R satisfies the restricted right minimum condition by [5]. Since condition c) isright-left symmetric, the equivalence of a) and b) followsimmediately. p A submodule U of the right R-module Qr is right invertible if there exist u1; ...; un 2 U and q1; ...; qn 2 R : U)` with u1q1 ...unqn 1 [11, Chaptersii.4 and IX.5]. Over an integral domain R,arightin- vertible module U hasthe additional property that UR : U)` R which may fail in the non-commutative setting: EXAMPLE 4.4. There exists a right and left Artinian, hereditary ring R such that Qr isa perfect right and left localization of R, which containsan r essential right ideal I with R : I)` Q and R : I)`I I. Moreover, there exist r1; r2; s1; s2 2 I and q1; q2; p1; p2 2 R : I)` satisfying 1 r1p1 r2p2 s1q1 s2q2 such that r1p1; r2p2 2 R, but s1q1; s2q2 62 R. Thus, IR : I)` 6 R. PROOF. Let F be a field of characteristic different from 2, and R be the lower triangular matrix ring over F. Clearly, R isa right and left Artinian ring. According to [8, Theorem 4.7], R isa right hereditary ring, which is also left hereditary by [5, Corollary 8.18]. Finally, by [8, Proposition 2.28 and Theorem 2.30], the maximal right and left ring of quotientsofR is 10 Q Mat F). Inside R, we consider the idempotents e and 2 1 00 00 F 0 e2 . Let I , a two-sided ideal of R which is essential as 01 F 0 a 0 00 00 a right ideal because for all a; b; c 2 F. Since bc 10 c 0 I Qre , we have R : I) Qr and R : I) I I. 1 ` ` 10 10 Finally, consider the elements s and s 1 À10 2 10 1 À 1 1 1 of I and q 2 2 and q 2 2 of Q. It iseasyto see 1 c 0 2 c 0 that s q ; s q 62 R although s q s q 1. On the other hand, 1 1 2 2 1 1 2 2 00 01 setting r p e 2 I, r 2 I,andp yields 1 1 1 2 10 2 00 r1p1; r2p2 2 R and r1p1 r2p2 1. p 56 Ulrich Albrecht r In view of the previousexample, we define a submodule U of QR to r be strongly invertible if there isa submodule M of RQ such that MU UM R. LEMMA 4.5. Let R be a right and left non-singular, right and left Utumi-ring with maximal right and left ring of quotients Q. A submodule UofQr is strongly invertible if and only if it satisfies the following con- ditions: i) U is also a submodule of RQ. ii) UR is a finitely generated projective generator of MR. iii) RU is a finitely generated projective generator of RM. PROOF. Suppose that U is strongly invertible, and choose a r submodule X of RQ with XU UX R.Then,X R : U)` \ R : U)r and R : U)`U R UR : U)r . Moreover, RU UX)U UXU) r UR U yieldsthat U isa submoduleof RQ too. By symmetry, X r is also a submodule of QR.Becauseofthis,R : U)` and R : U)r r r are submodules of both QR and RQ too. Therefore, UR : U)` UR : U)`R UR : U)`UR : U)r UR : U)r R.Bysymmetry, R : U)rU R. By what has been shown in the last paragraph, we can write 1 u1q1 ... unqn with u1; ...; un 2 U and q1; ...; qn 2 R : U)`. Let fi : U ! R be left multiplication by qi. Asin [11, Proposition IX.5.2], the set fu1; f1); ...; un; fn)g isa projective basisfor U. Select v1; ...; vm 2 U m and p1; ...; pm 2 R : U)` with 1 p1v1 ... pmvm. Define c : U ! R m m by cx1; ...; xm) Si1pixi . Then, c isonto, and U R È W, i.e. U isa generator of MR. By symmetry, RU isa finitely generated projective generator of RM. Conversely, assume that U satisfies the three conditions. Observe r r that R : U)` and R : U)r are submodules of both QR and RQ . Since it ` isa projective generator of MR,thereis` PROPOSITION 4.6. Let R be a right and left non-singular, right and left Utumi-ring. If I is a two-sided ideal of R such that RI and IR are finitely generated projective generators of RM and MR respectively, then R=I is projective with respect to all RD-exact sequences. PROOF. The proof of [7, LemmasI.7.2 and I.7.4] can be adapted to show that R=I isprojective with respectto all RD-exact sequencesof R-modulesprovided there are r1; ...; rn 2 R and q1; ...; qn 2 R : I)` such that r1q1 ... rnqn 1andr1q1; ...; rnqn 2 R. However, thisis guaranteed by Lemma 4.5. p Finally, the lattice of finitely generated right idealsover right strongly nonsingular, hereditary right and left Noetherian rings may not be dis- tributive: EXAMPLE 4.7. There exists a right strongly non-singular, hereditary, right and left Noetherian ring R for which the lattice of right idealsisnot distributive. PROOF. Let R be the ring considered in Example 4.4, whose notation Q 0 will be used in the following. 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