Embedding Rings with Krull Dimension in Artinian Rings

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1 Ž. JOUNAL OF ALGEBA 182, ATICLE NO Embeddng ngs wth Krull Dmenson n Artnan ngs Alfred Golde School of Mathematcs, Leeds Unersty, Leeds LS2 9JT, England and Gunter Krause Department of Mathematcs and Astronomy, Unersty of Mantoba, Wnnpeg, Mantoba 3T 2N2, Canada Communcated by A. W. Golde eceved August 22, INTODUCTION Any commutatve noetheran rng embeds as a subrng n an artnan rng, and for a long tme t was an open queston whether ths was also true for non-commutatve noetheran rngs. Whle such an artnan embeddng exsts for many classes of noetheran rngs, Dean and Stafford 2 showed that there are exceptons. They dd ths by provdng an example of a noetheran algebra over the feld of complex numbers that does not admt a fathful Sylvester rank functon, yet accordng to one of the man results n Schofeld s book 14, the exstence of such a functon s necessary and suffcent for an algebra over a feld to admt an embeddng nto an artnan rng, ndeed nto a smple artnan rng. On the postve sde, Blar and Small 1 used Schofeld s Theorem to prove that a rght noetheran k-algebra can be embedded n an artnan rng S such that S s flat, whenever contans no nonzero rght deal wth reduced rank zero. In 9 t was shown that for a noetheran k-algebra ths latter condton s necessary as well. If, furthermore, has fnte GelfandKrllov dmenson, then an artnan overrng can be obtaned by embeddng n a fnte Support by a research grant from NSEC s gratefully acknowledged. E-mal address: gkrause@cc.umantoba.ca $18.00 Copyrght 1996 by Academc Press, Inc. All rghts of reproducton n any form reserved. 534

2 EMBEDDING INGS IN ATINIAN INGS 535 drect sum of rreducble rngs, each of whch has an artnan quotent rng. As ths constructon does not make use of Schofeld s result, one naturally tres to carry t out when s a noetheran rng, but s no longer assumed to be an algebra over a feld. The man obstacle n dong so s that a symmetrc dmenson functon Ž such as the GelfandKrllov dmenson. s not avalable for noetheran rngs n general, makng t dffcult to establsh the ncomparablty condton for certan sets of prme deals. Thus, we are more or less forced to assume such a condton as one of the hypotheses. However, t turns out that the rng need not be noetheran, t s enough to assume that t has rght Krull dmenson. To date, no example has been found of a noetheran rng wth two comparable prme deals that belong to the same clque. Whether clques always satsfy the ncomparablty condton s a long-standng open queston Žsee, for example, 5, Problem 11, p for noetheran rngs. Thus, our result Ž Corollary 8. that a noetheran rng embeds n an artnan rng, provded t satsfes the ncomparablty condton for clques and has no nonzero rght deals of reduced rank zero, seems best possble under the crcumstances. 2. DEFINITIONS AND NOTATIONS All rngs consdered are assocatve wth unt element 1, modules are untary. For standard termnology the reader s referred to 5, 13. Let be a rng wth rght Krull dmenson KdmŽ.. The noton of the reduced rank, orgnally defned for fntely generated modules over noetheran rngs, can be extended to rght -modules wth Krull dmenson, as was shown by Lenagan n 11. Thus, s a functon that assgns to each rght -module M wth Krull dmenson a nonnegatve nteger Ž M., such that Ž M. Ž K. Ž L. for each short exact sequence 0 K ML0. ecall that Ž M. 0 f and only f for each m M there exsts an element c, regular modulo the nl radcal N of, such that mc 0. Assocated wth s a Gabrel flter Žsee, e.g., 15. F, consstng of all rght deals F of wth Ž F. 0. The -closure, orf-closure, or smply closure n M of a submodule N of a rght -module M s the submodule cl, MŽ N. clž N. m MmFNfor some F F 4. The submodule N s called -closed or closed n M f N clž N.. A rght -module M s called -torsonfree, or wthout -torson f clž Note that a rght -module M wth Krull dmenson satsfes the ascendng chan condton as well as the descendng chan condton for closed submodules.

3 536 GOLDIE AND KAUSE If X s a subset of the rng, and M and W are rght and left -modules, respectvely, then lmž X. annhlator of X n M m MmX0 4, rwž X. annhlator of X n W w WXw0 4. If X s a subset of the rght -module M, then r Ž X. s the rght annhlator of X n, and smlarly l Ž X. denotes ts left annhlator f M s a left -module. The subscrpts wll be deleted f there s no danger of ambguty. A prme deal P of s assocated wth the rght -module M f there exsts a submodule 0 N M such that P rž N. for all submodules 0 N N. AssŽ M. set of assocated prmes of the -module M SpecŽ. set of all prme deals of mnspecž. set of mnmal prmes of NNŽ. nl radcal of PSpecŽ. P EŽ M. njectve envelope of the module M. If I s an deal of the rng, then C Ž I. ccximples that x I 4, CŽ I. cxcimples that x I 4, CŽ I. CŽ I. C Ž I.. An afflated seres of a rght -module M s a sequence of submodules 0 M0 M1 Mn1 Mn M, together wth a set of prme deals P,...,P 4 1 n called afflated prmes such that each P s maxmal n AssŽ MM. and M M l Ž P. 1 1 MM. The set of all afflated prmes 1 of M, that s, the set of prme deals that appear n some afflated seres of M, s denoted by AffŽ M.. Let be a rng, and let Q, P SpecŽ.. Then Q s lnked to P Ž a A., denoted by Q P, f As an deal wth QP A Q P, such that Q PA s torsonfree as a rght P-module Žthat s, no nonzero element s annhlated by an element n CŽ P.. and fully fathful Žthat s, has no nonzero unfathful submodules. as a left Q-module. The graph of lnks or the lnk graph of s the drected graph whose vertces are the elements of SpecŽ., wth an arrow from Q to P whenever Q P. The connected components of ths graph are called clques. For P SpecŽ., the Ž unque. clque contanng P s denoted by ClqŽ P..

4 EMBEDDING INGS IN ATINIAN INGS 537 A subset X of SpecŽ. s sad to be rght lnk closed f whenever P X and Q P, t follows that Q X. The rght lnk closure r Ž X. of a set X of prmes s the smallest rght lnk closed subset of SpecŽ. contanng X. Note that for a prme deal P, r Ž P. conssts of all prmes Q SpecŽ. for whch there s a sequence of prme deals Q Q 1, Q 2,...,QnP wth Q Q for 1,...,n1. Obvously, r Ž P. ClqŽ P. 1. A set X of prme deals s sad to satsfy the ncomparablty condton f P Q for P, Q X mples that P Q. 3. -TOSIONFEE MODULES Let be a rng wth rght Krull dmenson, and let M be a nonzero rght -module. By 6, Theorem 7.1, satsfes the maxmum condton for prme deals, and AssŽ M. by 6, Theorem 8.3. Thus, one can attempt to construct an afflated seres for M n the usual way, but the constructon may not reach the top n fntely many steps. Our frst proposton shows that no problem arses for the case of an njectve rght -module E wth cl Ž., E 0 0, snce t turns out that E has fnte length as a module over ts endomorphsm rng. Note that ths latter fact follows from a general result about -modules: A quas-njectve rght -module M Ž where s any rng. s a -module Žthat s, has DCC for annhlators of subsets of M., f and only f M has fnte length as a module over End Ž M. Žsee, e.g., 3, Proposton 8.1, p Note also that for the case EE Ž., Proposton 1 below specalzes to Theorem 2.2 of 4. POPOSITION 1. Let be a rng wth rght Krull dmenson, let E be an njecte rght -module, let H End Ž E., and let J denote the Jacobson radcal of H. Then the followng are equalent. Ž. cl Ž 0., E 0. Ž. E has fnte length Ž., and AssŽ E. mnspecž. H. Ž. E s noetheran, and AssŽ E. mnspecž.. H If any of the aboe hold, then J s nlpotent of nlpotency ndex. Furthermore, f E has fnte unform dmenson, then HJ s semsmple artnan, so H s semprmary. Proof. Note that Ž., so the addtvty of the reduced rank mples the ascendng and descendng chan condtons for -closed rght deals.

5 538 GOLDIE AND KAUSE Ž. Ž.. Let N be a submodule of H E. Snce annhlators n of subsets of N are closed rght deals of, t follows that r Ž N. rž x. rž x. xn for fntely many elements x N. Gven any element y l Žr Ž N.. E, the -homomorphsm : Ž x, x,..., x. ryr, r, from 1 2 n n 1 Ž x, x,..., x. x x x E n 1 2 n 1 2 n Ž n. n nto E can be extended to an element of Hom E, E H. Thus, there exst h 1, h 2,...,hnH wth Ž. Ž h x h x h x. r Ž x, x,..., x. r yr, n n 1 2 n for all r. In partcular, y h x h x h x, provng that l Žr Ž N n n E Hx1 Hxn N. Thus, f N M HE, then the closed rght deal r Ž N. properly contans the closed rght deal r Ž M., so any ascendng or descendng chan of submodules of E can have at most Ž. H proper nclusons. It s clear that any P AssŽ E. must be a mnmal prme, snce non-mnmal prmes are n F. Ž. Ž.. Ths s trval. Ž. Ž.. Assume that K cl Ž 0., E 0. Note that K s an H-sub-bmodule of E. Snce K s fntely generated, r Ž K. H H F. Snce r Ž K. P for some P AssŽ E., ths would mply that a mnmal prme belongs to F, whch s mpossble. It s clear that J E 0, where denotes the composton length of H E. Fnally, f E has fnte unform dmenson, then t s well known that HJ s semsmple artnan Žsee, for example, 10, Proposton 2, p In 9, Theorem 3.5, t was shown that for a rght noetheran rng whose rght deals are fntely annhlated, the condton cl Ž., 0 0s equvalent to AssŽ. mnspecž.. Usng the above proposton, ths can be generalzed as follows. COOLLAY 2. Let be a rng wth rght Krull dmenson. Then the followng statements are equalent. Ž. AssŽ. mnspecž., and rght deals of are fntely annhlated. Ž. cl Ž 0. 0.,

6 EMBEDDING INGS IN ATINIAN INGS 539 Proof. Ž. Ž.. Assume that K cl Ž 0. 0, so rž K.,, beng the ntersecton of fntely many rght deals n F, would be n F. As above, ths s mpossble, because t would gve a mnmal prme deal n F. Ž. Ž.. Let E EŽ., H End Ž E., and note that cl Ž 0., E 0, as s essental n E. By Proposton 1, AssŽ E. mnspecž., and H E s noetheran. Thus, for any rght deal A of, HA Ha Ha, a A. 1 n Ž. Ž. Ž. Ž. Hence r A r HA r a r a,so As fntely annhlated. 1 n As a consequence of Proposton 1, gven a rng wth rght Krull dmenson, a -torsonfree njectve rght -module E wth endomorphsm rng H has an afflated seres, whose members, beng H--bmodules, have fnte length as left H-modules. Although we shall not make use of ths fact later on, t turns out that the sets of prmes arsng from any two afflated seres of E are dentcal. Ths s a consequence of the followng lemma, whch s a slght generalzaton of 5, Lemma LEMMA 3. Let be a rng wth rght Krull dmenson, let HB be a bmodule such that H B has fnte length, and let 0 V V V V V V B n1 n be a sequence of sub-bmodules wth P rvv Ž. AssŽ V V. 1 1 for all 1n. Then P,...,P 4 AffŽ B.. 1 n Proof. Let 0 W0 Wj1 Wj Wm B be a bmodule composton seres of B wth Q rww Ž. AssŽ W W. j j j1 j j1. By the JordanHolder Theorem, t suffces to show that P,..., P4 1 n Q,...,Q 4 1 m, and t may be assumed that the composton seres s a refnement of the seres of the V s. Furthermore, we may reduce to the case n 1. Obvously, P rv Ž. rž B. rw Ž. 1 1 j Qj for all j. Assume that P Q for some j, and let c Q CŽ P. 1 j j 1. Snce HB s noetheran, B s torsonfree as an P1-module, so rght multplcaton by c nduces an H-somorphsm, whence Wj WcW j j1, whch s mpossble n a module of fnte length. The next result s the key for obtanng an embeddng of a -torsonfree rng wth rght Krull dmenson nto a rght artnan rng, because t descrbes the set of regular elements of n terms of the afflated prmes of EŽ.. THEOEM 4. Let be a rng wth rght Krull dmenson, and let HB be a bmodule such that B has fnte length and cl Ž Let H, B 0 V V V V V V B n1 n

7 540 GOLDIE AND KAUSE be a sequence of sub-bmodules wth P rvv Ž. AssŽ V V. 1 1 for all 1n. Then Ž. Ž VV. cvv for any c CŽ P. 1 1, 1,...,n. Ž. VcV for any c CŽ P. j1 j, 1,...,n. Ž. Ž Ž.. C r V CŽ P., 1,...,n. j1 j Proof. Ž. Set X VV 1, and observe that as an P-module X s torsonfree, snce H X s fntely generated and snce P s the unque assocated prme of X. Thus, for any c CŽ P., the map x xc, x X, s a left H-module somorphsm from X onto XcX,so XcX,as H X s artnan. Ž. Proceed by nducton on, the case 1 beng a consequence of Ž. 1. Assume that the statement s true for 1, and let c CŽ P. j1 j. Then Ž. by, so V1 V1 V1cV c V V V V V cv V cvcv c Ž. Let c CŽrV Ž... Then as rght rž V-modules. and hence also as rght -modules crž V. crž V., rž V. rž V. rž V. whence Žc rv Ž.. 0, and consequently also Žc rv Ž.. j 0 for any j, as rv Ž. rv. Ž. j Snce B s -torsonfree, the map c, V, j s a left H-module somorphsm from Vj onto VcV j j for any j. Snce HVj s artnan, VcV j j follows. Assume now that cx Pj for some x. Then Vj VjcVj1 Vj 0 cx x x, Vj1 Vj1 Vj1 Ž. Ž Ž.. so x P. Thus c C P for all j, whence C rv CP. Ž. j j j1 j For the reverse ncluson, let c CŽ P. j1 j, and assume that Vcx0 for some x. As VcV by Ž., t follows that x rv, Ž. hence c C ŽrV Ž... To see that c CŽrV Ž.. as well, assume that Vxc0 for some x rv. Ž. Then VxVV j j1 for some 1 j. Snce Ž VxV V. j1 j1 c0, ths contradcts the fact that VjVj1 has no CŽ P. -torson elements. j

8 EMBEDDING INGS IN ATINIAN INGS ATINIAN EMBEDDINGS The purpose of ths secton s to prove that a rng wth rght Krull dmenson and cl Ž., 0 0 embeds n a rght artnan rng, provded that the rght lnk closure of each P AssŽ. satsfes the ncomparablty condton. Ths wll be acheved by dentfyng a fnte number of factor rngs of that have artnan quotent rngs, and by subsequently embeddng nto a drect sum of these quotent rngs. We begn wth a generalzaton of Jategaonkar s Man Lemma 7, 6.1.3, adaptng a proof gven by Lenagan and Warfeld 12, Theorem 1.2 for modules over noetheran rngs. POPOSITION 5. Let be a rng wth rght Krull dmenson, and let H M be a bmodule that satsfes the descendng chan condton for sub-bmodules. Assume that M has an afflated seres 0 M M M M M M n1 n wth afflated prmes P rž MM. 1, such that each MM1 s torson- free as a rght P-module. Let be the least nteger such that there exsts a submodule M of M wth M M and M n1 Pn M. Then Ž. If 0, then P ržm. n. Ž. If 0, then P P. n Proof. Note that each M s an H--sub-bmodule of M. Proceed by nducton on n, the result beng trval for n 1. Let n 1, and assume that the result holds for any H--bmodule wth DCC on sub-bmodules that has an afflated seres of the above knd of length n. If 0, then P r M r M n Ž. Ž Mn1Mn1. P n, so ržm. P n. If 1, then, usng the nductve hypothess on the afflated seres 0 M M M M M M MM n1 1 1 of MM1 of length n 1, t follows that Pn P. Ths leaves the case when 1, so 0 M P M. Let X be a rght -submodule of M n 1, such that HX s mnmal n the set HN N M, N M 4 n1. eplacng M by X, f needed, t may thus be assumed that rž N. ržm. for all submodules N of M wth N M. Set A rž M., and observe that n1 A r M r M M M r M M P P. Ž. Ž. Ž. n1 n1 1 n 1

9 542 GOLDIE AND KAUSE Assume that A P P. Now P P, for otherwse A P,so M 1 n 1 n 1 l Ž P. M 1 M1M n1, a contradcton. Consequently, MP1M n1, hence Ž. Ž. r M P r M. However, as MPP M Ž P P. 1 1 n 1 n MA0, ths contradcts M Pn 0. Thus PPAPP, n 1 n 1 and we proceed to show that P P va A. n 1 If P P A s not fully fathful as a left P -module, then there n 1 n exsts an deal B wth A B P P and an deal X P such n 1 n that XB A, so M XB 0. Now, M X M n1, for otherwse X Ž. Ž r M M M P. Thus A r M. ržm X. B, whch gves n1 n1 n a contradcton. If Pn P1A s not torsonfree as a rght P1-module, then there exsts an element b A, b P P, and c CŽ P., such that bc A, n 1 1 so M bc 0. As 0 M b M Pn M 1, ths contradcts the hypothess that M s torsonfree as a rght P -module. 1 1 emark. It s obvous from the proof that the above proposton remans true, f the descendng chan condton for H--sub-bmodules of M s replaced by the condton that satsfes the maxmum condton for annhlators of subfactors of M. Thus t holds, for example, f s assumed to be rght noetheran. If s noetheran, then, furthermore, t need not be assumed that each M M s torsonfree as a rght P - 1 Ž. module see 12, Theorem 1.2. THEOEM 6. Let be a rng wth rght Krull dmenson, and let HB be a bmodule such that B has fnte length and cl Ž. H, B 0 0. Then Ž. AffŽ B. r ŽAssŽ B... Ž. r Ž B. has a rght artnan, classcal rght quotent rng f and only f r ŽAssŽ B.. satsfes the ncomparablty condton. In ths case, AffŽ B. r ŽAssŽ B.. PPr Ž B. mnspecžrž B..4. Proof. Gven any afflated seres 0 V V V V V B n of B wth afflated prmes P rvv Ž. 1, each VV1 s torsonfree as a rght P-module, so Proposton 5 can be appled to each of the H--bmodules V. Snce P AssŽ B., trvally P r ŽAssŽ B Assume that for 1 t has been establshed that P,...,P 4 r ŽAssŽ B By Ž Proposton 5, ether P r M. for some submodule M V1 or P Pj for some j. In the frst case, P Q for some Q AssŽ B., and hence P Q, as AssŽ B. mnspecž., due to the hypothess that cl Ž 0., B 0, so P AssŽ B. r ŽAssŽ B... In the second case, P r ŽAssŽ B.. by the nductve hypothess. Thus Ž. has been establshed.

10 EMBEDDING INGS IN ATINIAN INGS 543 For Ž., assume wthout loss of generalty that r Ž B. 0. Takng the above afflated seres for B, t follows that BP P P 0, so that n n1 1 r mnspecž. P,...,P 4 AffŽ B. Ž AssŽ B.., 1 n so equalty holds throughout, f the ncomparablty condton s assumed Ž. n for the latter set. Thus C N CŽ P. 1. By Theorem 4, the second set equals CŽ. 0, so CŽ N. C0 Ž., and has a rght artnan, classcal rght quotent rng by 13, Theorem For the converse, recall that AssŽ B. mnspecž., so n order to establsh the ncomparablty condr ton for ŽAssŽ B.. t s suffcent to show that f Q P, and f P s a mnmal prme, then Q s also mnmal. Assume that the lnk s va A, and that Q s not mnmal. Then Q contans an element c CŽ N. CŽ 0.. Snce cqp Ž. AQP, and snce cqp Ž. QPas rght - modules, t follows that ŽQ PcŽ QP.. 0, and hence that ŽQ PA. 0. As Q PA has no CŽ P. -torson, hence no CŽ N.-torson, ths s a contradcton. COOLLAY 7. Let be a rng wth rght Krull dmenson such that cl Ž If r Ž P., satsfes the ncomparablty condton for eery PAssŽ., then embeds n a rght artnan rng. Proof. Let E E Ž., and let AssŽ E. AssŽ. P, P,...,P n. Then E E1 E2 E n, each E beng an njectve rght -module wth AssŽ E. P 4. Now, cl Ž., E 0 0 and E has fnte length over H End Ž E. by Proposton 1. Snce r ŽAssŽ E.. r Ž P. s assumed to satsfy the ncomparablty condton, each of the rngs r Ž E. has a rght artnan rght quotent rng by the precedng theorem. The result n follows, snce r Ž E. r Ž E. r Ž For many classes of noetheran rngs the ncomparablty condton has been establshed for clques. Ths s usually done usng a symmetrc dmenson functon avalable for the partcular class, for example, the GelfandKrllov dmenson Ž provded t s fnte. n the case of noetheran algebras, or the classcal Krull dmenson n the case of noetheran rngs satsfyng the second layer condton Žsee, e.g., 5, Corollary To date, no example s known of a noetheran rng wth a clque that contans comparable prmes, so the followng corollary seems qute general. COOLLAY 8. Let be a noetheran rng wth cl Ž If ClqŽ P., satsfes the ncomparablty condton for each P AssŽ., then s a subrng of an artnan rng. Proof. Observe that r Ž P. ClqŽ P. for each P AssŽ., and decompose EŽ. as n the proof of the precedng corollary. The clam now

11 544 GOLDIE AND KAUSE follows from Theorem 6, takng nto account that for a noetheran rng CŽ N. CŽ 0. mples the exstence of a two-sded artnan quotent rng. In general, one cannot expect the embeddablty of a noetheran rng nto an artnan rng S to force to be -torsonfree. However, ths does happen when S s flat, as s the case when S s an artnan quotent rng of. POPOSITION 9. Let be a rght noetheran subrng of a rght artnan rng S such that S s flat. Then cl Ž., 0 0. Proof. Defnng Ž M. lengthž M S. for a fntely generated rght -module M produces an addtve rank functon. Let A be a rght deal of wth Ž A. 0. Then Ž A. 0 by 8, Lemma 1.3, hence A S 0. As S s flat, A S AS canoncally, so A 0. However, the followng example shows that a noetheran rng may have -torson even when t s close to havng an artnan quotent rng, n the sense that there are deals I j,1jn, wth n j1ij 0, such that each of the rngs Ij has an artnan quotent rng. EXAMPLE. Let Z denote the rng of ntegers, let Zxbe the polynomal rng n one varable x, and let M ZxŽ x., vewed as a Z-Zx- Zx 0 Ž x. bmodule. In the rng ž /, I I 0, where I ž / and M Z Z 0 I. Note that I and I Zx 2 ž / 1 ž / 2, so both I1 and M Z Z Z I2 have artnan classcal quotent rngs. Now, consder the prme deals Ž x P Q, and note that P 0. Thus cl Ž ž M Z/ ž M Z/ ž M 0/, EFEENCES 1. W. D. Blar and L. W. Small, Embeddngs n artnan rngs and Sylvester rank functons, Israel J. Math. 58 Ž 1987., C. Dean and J. T. Stafford, A nonembeddable noetheran rng, J. Algebra 115 Ž 1988., C. Fath, Injectve Modules and Injectve Quotent ngs, Lecture Notes n Pure and Appled Mathematcs, Vol. 72, Dekker, New York, A. W. Golde, ngs wth an addtve rank functon, J. Algebra 133 Ž 1990., K.. Goodearl and. B. Warfeld, Jr., An Introducton to Noncommutatve Noetheran ngs, London Math. Soc. Stud. Texts, Vol. 16, Cambrdge Unv. Press, Cambrdge, UK, Gordon and J. C. obson, Krull dmenson, Mem. Amer. Math. Soc. 133 Ž A. V. Jategaonkar, Localzaton n Noetheran ngs, London Math. Soc. Lecture Notes, Vol. 98, Cambrdge Unv. Press, Cambrdge, UK, G. Krause, Addtve rank functons n noetheran rngs, J. Algebra 130 Ž 1990., G. Krause, Flat embeddngs of noetheran algebras n artnan rngs, Israel J. Math. 77 Ž 1992.,

12 EMBEDDING INGS IN ATINIAN INGS J. Lambek, Lectures on ngs and Modules, Gnn Ž Blasdell., Waltham, MA, T. H. Lenagan, educed rank n rngs wth Krull dmenson, n ng Theory ŽProv. Antwerp Conf. Ž NATO Adv. Study Inst.., Unv. Antwerp, Antwerp, 1978., pp , Lecture Notes n Pure and Appl. Math., Vol. 51, Dekker, New York, T. H. Lenagan and. B. Warfeld, Jr., Afflated seres and extensons of modules, J. Algebra 142 Ž 1991., J. C. McConnell and J. C. obson, Noncommutatve Noetheran ngs, Wley, New York, A. H. Schofeld, epresentatons of ngs over Skew Felds, London Math. Soc. Lecture Notes, Vol. 92, Cambrdge Unv. Press, Cambrdge, UK, B. Stenstrom, ngs of Quotents, Sprnger-Verlag, BerlnNew York, 1975.